Slow-Roll: What is the energy of a scalar field transformed to due to Hubble friction? So, I've lately been reading about the Slow-Roll approximation of scalar fields. However - since I don't really know that much about cosmology - I have some trouble regarding the understanding of the Hubble friction term. If I understand it correctly, the scalar field dissipates energy due to Hubble friction which is caused by the expansion of the universe. But what I'm wondering about now is since there should be energy conservation: What is the energy being transformed to if the scalar field "loses" it?
 A: As already explained in other answers and comments in General Relativity (GR) energy is not conserved. Some people and physicists say it is, it simply gets lost by the matter-energy and gained by the gravitational field, or viceversa; this is more pleasing to our sense of conservation of something, but it has problems in that the gravitational energy, is not defined covariantly, so it can change in different coordinate systems. The caveat is that you still can define a conserved energy when there is time symmetry (in GR, a timelike Killing vector), and you can define the total energy (but not energy density at each point in spacetime) for a spacetime that is asymptotically flat (the Bondi and ADM energies, ADM includes the gravitational energy, Bondi separates out that of gravitational waves). Since for most practical purposes you can approximate an isolated object as being in an asymptotically flat spacetime, you can calculate. 
The phi dot term you ask about involves a dissipation of energy term that is in fact like that. As phi goes down its potential energy slope it looses energy and provides it to the universe's expansion. The meaning of Hubble friction is that it goes slower than it would without it, and it does that because the universe's fast expansion slows down the roll down. Like any friction, it slows things down: in the cases of the scalar field you wrote about it slows the fast roll it would normally do in a potential valley (down it to the minimum) to a slow roll, making the inflation (predicated on that scale field) take the amount of time needed to expand the universe by about 60 e foldings, as needed to explain its size and uniformity, which is what inflation explains. This was the change made by Linde as 'new inflation', that made inflation theory consistent. You could say it gave that energy up to gravity by expanding the universe, but that's what GR purists would rather you did not say, and you would just say that the the process created energy in the universe's expansion. It matters less the exact words, more that the math and predictions are the same either way. The fact is that it works ok in the equations if phi does what you have, and the universe's size increases exponentially. The friction is a little of a misnomer, it does slow the roll down, but as phi goes down the curve of potential energy to its minimum it gives up that energy towards the expansion. 
In cosmology there are other cases of energy being gained or lost. Note that the cosmological spacetime is described, in the big, by the Roberston Walker metric, and it is not time symmetric, nor asymptotically flat, so energy is not (in the purist sense) conserved. Other cases: in the expansion of the universe the photons that make up the cosmic microwave background incur a red shift, where there are (roughly) the same number of photons, but each has a lower frequency, and this less energy. They lost energy. Similarly for any other radiative field. In Newtonian mechanics we'd say the gravitational potential energy picked up the kinetic energy given up by the photons. In GR the purists say energy was lost, others may say the gravitational field picked up the energy. Similarly for photons radiated as Hawking radiation from near a black hole, as they travel further away they get more and more red shifted, loose energy.
The important points are: careful with energy in GR, though it can be a physically powerful help also; and the scalar field as it slow rolls down its potential slope seems to be the best explanation we have for inflation and the symmetries of the observed universe. 
