What would happen if I took empty space and stretched it? There was a talk at my school by Rocky Kolb and he claimed that they derived (or it might have been ''experimentally found'', I don't remember) that the mass of empty space must be/is on the order of $10^{-30}g/cm^3$. This made me curious - what would happen if I somehow took a unit volume of empty space (cube) and pulled on it from all sides, increasing its volume by $dV$?
I'm curious as to what that would mean with regards to the new volume of space $V+dV$ and what that would mean to the space around it and the rest of the universe? Do we even have ideas/theories as to what changes as you change empty space (I explicitly ask this because we are probably nowhere near having the ability to modify space itself)?
 A: Space expands as described by the standard cosmology model, Lambda CDM, using the FLRW spacetime metric, and due to the Big Bang which started the expansion, the very early inflation (cause not to totally agreed but the standard is some form of inflation TBD field), the radiation and matter densities as it evolves, and as we found out in the last 20 years the expansion is accelerating because the dark energy (the cosmological constant) is positive and is repulsive. It has indeed approx the mass-energy density you stated. See the Wiki article on it, it summarizes what is known about it, and the evidence for it. https://en.m.wikipedia.org/wiki/Dark_energy. 
It constitutes about 68% of the mass energy of the universe - for the past few billion years it has been more than the matter content of the universe. The competing theory to the cosmological constant as a source is quintessence, but it has not had much luck with data to back it up. 
It is in some sense the energy density of vacuum spacetime, but other than the equations used to estimate its e effects on the metric (spacetime), little is known. It may be the quantum zero,point energy of the vacuum. Either way, we know what it is. 
And yes, it is a constant density, and as space (the spatial slices at any value of the cosmological time in the spacetime, it is quite well defined) expands the total mass energy increases. In the cosmological model it keep increasing for hundreds of billions of years as space expands, with matter density getting closer to zero, it becomes so diffuse that it's all dark energy. 
If you could expand large regions of space (large means volumes comparable to what becomes cosmologically significant), a minimum of about 300 light years in extension, to bigger sizes, yes you'd create more dark energy (and you probably will have to spend that much energy or more to stretch it unnaturally). Smaller region like our solar system is just too small to affect it much. But we don't know how to do anything to space, or spacetime, the metric reacts to the metric around it (a nonlinear process), and to the stress energy tensor and the cosmological constant, so you'd have to arrange thing just so to have it do what you want. And how do you arrange such large quantities of energy? You don't. Spacetime has done it, it is expanding and creating more dark density as it does. 
So yes, if you magically stretches spacetime where spatial volume increases, and it is done on a large enough scale, yes, you can have more dark energy around. But we don't know how to use the dark energy we already have, so what for?
Maybe when we discover the right theory of quantum gravity we should then know something about what dark energy is. 
****EDIT from @freecharly's comment below****
He correctly points out a new set of analysis done on a larger set of supernova data indicating the acceleration is only supported by 3 sigma data, as opposed to the 5 sigma as the standard. Possibly indicating there is no acceleration and thus no dark energy. 
That article is a very incomplete analysis of all the observations that led to the conclusion that it is accelerating, that there is dark energy, and that led to the Nobel prize for the discovery. If you add up all the evidence it is still overwhelming. And there's some ar time to also surfaced that the statistical analysis they did did not weight the different supernova data the right way. 
Answer to Freecharly: my sense is to doubt the conclusion. I have not examined the full details but it is known that the dark energy and accelerating expansion is not just evidenced by the supernova data. There is also the CMB data and data on the matter density (which does not account for about 70% of the mass-energy of the universe), along with consistency with general relativity. When you add them all up its pretty overwhelming. the authors only used their data on supernova, and may have not weighted the different supernova data (further, closer, etc) statistically the way it probably should be. You get closer to 4 sigma for the supernova data. 
See the Forbes (which is not definitive but an indication of what other astronomers, astrophysicists and cosmologists, may be thinking, and the reasons, and MORE IMPORTANTLY the combined evidence for all 3 sets of evidence) article with some graphics on the combination of evidence. And stay posted, I'm sure there will be more analysis by others and more definitive conclusions (i.e., if it really is an issue or not)
http://www.forbes.com/sites/startswithabang/2016/10/27/new-supernova-results-is-the-universe-not-accelerating/#7e9c6f4667ca
A: Space is not an "object" that you can hold it from sides and make it stretch.
There are many perspectives to consider while we are calculating the energy of the empty space (vacuum energy). This is important cause $10^{-30}$ is not the only answer that someone can give to you. 
1-Cosmological perspective, observations suggest that the energy density of the empty space should be $7\times \,10^{-27}kg/m^3$ (We can measure it by observations such as WMAP) 
This energy can be expressed as  cosmological constant. The interesting part of the cosmological constant is, the energy density does not change with the expansion of the universe. For instance, take a box with volume of $a^3$. And assume that, the matter density of the box is $\rho_m$ , radiation density is $\rho_r$ and there is Lambda density $\rho_{\Lambda}$. 
Let us increase the size of the box to $2a$ so that the volume of the box becomes ${8a^3}$. 
The matter density will become $\rho_m/8$, radiation density will become  $\rho_m/16 $ but the energy density of cosmological constant( $\rho_{\Lambda}$ )will not change. So the  energy density of cosmological constant does not depend on the size of the box (the volume that you choose in space).
2-  Quantum Field theory perspective, 
Well there are actually 4 different views on the subject. I find a site that explains this. 

2.a)We can try to calculate the energy density of the vacuum using quantum field theory. If we calculate the lowest possible energy of a harmonic oscillator, we get a bigger answer when we use quantum mechanics than when we use classical mechanics. The difference is called the "zero-point energy". The zero-point energy of a harmonic oscillator is $1/2$ Planck's constant times its frequency. Naively we can try calculating the energy density of the vacuum by simply summing up the zero-point energies of all the vibrational modes of the quantum fields we are considering (e.g. the electromagnetic field and various other fields for other forces and particles). Vibrational modes with shorter wavelengths have higher frequencies and contribute more vacuum energy density. If we assume spacetime is a continuum, we have modes with arbitrarily short wavelengths, so we get INFINITY as the vacuum energy density. But there are problems with this calculation....
2.b)Slightly less naive way to calculate the vacuum energy in quantum field theory is to admit that we don't know spacetime is a continuum, and only sum the zero-point energies for vibrational modes having wavelengths bigger than, say, the Planck length (about 10-35 meters). This gives an ENORMOUS BUT FINITE vacuum energy density. Using $E = mc2$ to convert between energy and mass, it corresponds to a mass density of about $10^96$ kilograms per cubic meter! But there are problems with this calculation, too....
  One problem is that treating the vibrational modes of our fields as harmonic oscillators is only valid for "free field theories" - those in which there are no interactions between modes. This is not physically realistic.
  However, while taking interactions into account changes the precise answer, we are still left with an enormous energy density. The ridiculous ratio between this density and what's actually observed is often called the cosmological constant problem. One way to put it is that in units of Planck mass per Planck length cubed, the cosmological constant is about $10^{-123}$. It's hard to make up a theory that explains such a tiny nonzero number.
  But there's an even bigger problem, too....
2.c)Quantum field theory as it is ordinarily done ignores gravity. But as long as one is ignoring gravity, one can add any constant to ones definition of energy density without changing the predictions for anything you can experimentally measure. The reason is that without measuring the curvature of spacetime, one can only measure energy differences. The big problem with calculations 2 and 3 is that they ignore this fact. If we take advantage of this fact we are free to redefine energy density by subtracting off the zero-point energy, leaving an energy density of ZERO. In fact this is what is ordinarily done in quantum field theory.
2.d)An even less naive way to think about the vacuum energy density in quantum field theory is the following. In quantum field theory we are neglecting gravity. This means we are free to add any constant whatsoever to our definition of energy density. As long as we are free to do this, we can't really say what the vacuum energy density "really is". In other words, if we only consider quantum field theory and not general relativity, the vacuum energy density is NOT DETERMINED.

Reference
So the energy of the vacuum changes from which perspective you look. And its one of the unsolved problems. 
  
A: As the previous answers in here described very well, expanding space in volume non-intuitively keeps its mass density fixed and therefore appears "magically" to increase its total energy-mass when expanded, phenomenon we call Dark Energy described also as cosmological constant problem.
From where this extra energy-mass sources in order to make its mass density independent of its volume no one really seems to understand. A new "Einstein" must emerge to explain it.
There are today many theories about this but there is no consensus and all lack the rigorous phenomenology to put these theories in test.
Our inability to grasp the vacuum problem and what is called one of the biggest failures of modern physics is the discrepancy of the 120 orders of magnitude between the predicted value of the cosmological constant (dark energy) and the actual very small measured value you mentioned about $10^{-30}g/cm^3$ and called the Vacuum Catastrophe.
It seems that most of the part of the vacuum space energy is "hidden" to us and totally inaccessible to perceive with our apparatus.
This might be because it could be an unknown phase of energy-mass even more different than a Bose-Einstein Condensate (Logaritmic BEC Vacuum Theory) or even a type of superfluid (SVT) and that we barely measuring and scratching its surface that what is called as Quantum Foam.
It is even suggested,  that the part of the energy that is apparently missing, is in a superluminal phase state within our Universe and that space-time of our Universe has two discrete phases. One, the luminous and sub-luminous constituting light and ordinary matter accordingly and the other is in the form of an unknown superluminous energy perceived by us as "empty" space. No means for direct observation is possible although there are clues and phenomena all over for this.
The problem will most of these theories is that there are impossible today to experimentally verify.
There is actually a very nice presentation about this problem by Physics 2004 Nobel laureate Frank Wilczek - Materiality of a Vacuum:
https://www.youtube.com/watch?v=TD9PofbrrLc (part 1)
https://www.youtube.com/watch?v=X7rxlCxSqw8 (part 2)
One is for sure. Vacuum space is not empty. There is much more there inside that we have ever imagined.
