Energy extraction from universe expansion? I recently learned that energy is not conserved according to general relativity. Inspired, I went looking for a way to harness this interesting feature of the theory and came across an idea that involves extracting energy from the universe's expansion. The idea is to attach a very long rope to a high mass object. As the universe expands the rope undergoes tension and can be used to pull something from which energy can be extracted such as an axle. Some of the extracted energy is then used to make more rope which is then attached to the previous stretch of rope so that more energy can be extracted.
One of the glaring assumptions that I notice is that the extracted energy per length pulled is greater than the energy required to make a stretch of rope of that length. This assumption seems to me more of an engineering problem than something that might violate the fundamental laws of physics.
Another assumption is that the rope will not break under the tension. I imagine that some of the mass at the end of the rope could be released gradually so that the tension is maintained at a constant value as the rope grows in length.
Does this idea violate the laws of physics as is known today? The idea seems somewhat unsettling to me as it seems to imply something coming from nothing (free energy), but then again the thought of a universe coming into being seems just as absurd to me.
 A: 
I recently learned that energy is not conserved according to general relativity.

That is not true, energy is conserved in general relativity. But

*

*gravitational energy is non-local, which means that we cannot assign a specific energy density to  a specific point  (but we can formulate energy conservation law for a finite region).


*specific form of energy conservation law depends on a boundary conditions on a region for which this law is written, which formalism is used for the description of gravitational field, which spacetime is taken as a reference …
The closest analogy is thermodynamics, which does not have the energy but rather multiple notions of energy: internal, enthalpy, Gibbs, etc. each concept having its uses. The same is true for GR: there are multiple definitions of energy for curved spacetimes with different versions geared toward specific problems.
So, just from knowing that some energy conservation law in the cosmological setting exists we must conclude that universe expansion is not a magical source of free energy but rather a reservoir from which we can extract energy (yes!), but this reservoir would be finite either on a fundamental level (e.g. imposed by cosmological horizon) or on a practical one (with limits imposed by speed of light/maximum achievable acceleration/minimum density of a cable etc.).
One thing to keep in mind, is that extraction of energy from cosmic expansion necessitates distances much larger than the size of largest gravitationaly bound systems (galactic clusters and superclusters), so we are talking about hundreds on megaparsecs distances and billions of years time intervals while mass of a galaxy would be considered small in such discussions.
Some examples:
Milne universe This is a cosmological model described just by special relativity without the use of curved spacetime machinery. It is a world (which can be identified with the insides of Minkowski spacetime light-cone of a single point, the Big Bang) filled with matter (we can think of it as an infinite number of point-like galaxies) undergoing homogeneous and isotropic expansion. Relativistic length contraction and time dilation ensures that the spatial geometry of the  constant proper time hypersurfaces is hyperbolic and the average mass density is constant on it.
In the nearest neighborhood of a given observer the Hubble law gives us the velocities of neighboring galaxies: $v=H(t) r$ (as long as $v\ll c$). Assuming that we stretch a cable to a nearby galaxy of mass $m$ and make the other end do some useful work (e.g. by turning generator's axle) we can extract the energy $m v^2/2$ if we slow this galaxy to a stop. In this model “the energy of cosmic expansion” is just the kinetic energy of galactic motion relative to the observer at origin. Once the galaxy is stopped there is no more energy to extract from it, for further energy we must start slowing down other galaxies (Of course, we could also extract resources from this stopped galaxy).
If the distances are large enough that the velocities of expansion are comparable with the speed of light we run into some trouble. While the extractable kinetic energy could greatly exceed the rest mass of the galaxy, we must catch up to this galaxy first, in order to attach a cable to it. So while the reservoir of energy available for extraction is infinite here, the extraction of energy beyond the small non-relativistic neighborhood involves chasing down galaxies that are running away from us at relativistic velocities, which would be more and more difficult and would take more and more time the closer the velocities are to the speed of light.
In a more general case of spatially infinite universe that does not have a cosmological horizon  the extraction of energy from cosmological expansion would have the same qualitative features as in the Milne model: the pool of energy is infinite, the energy that could be extracted from a single galaxy is always finite and the energy extraction from the more distant galaxies is subject to diminishing returns: we must travel further and for a longer time with the gains per unit of our efforts (however we measure them) steadily decreasing.
De Sitter universe This is solution of Einstein equations with positive cosmological constant that possesses cosmological horizon.
Let us consider the situation along the lines suggested by OP: a small mass is tethered by a cable to a larger central mass which we consider  the  center of our reference frame (and of de Sitter static patch). If we vary the cable length slowly (so that configuration could be considered quasi-static), then ignoring possible radiative losses the total Killing energy of the system remains constant. At the same time the Killing energy of a mass $m$ held static in position with the radial coordinate $r$ is:
$$
E_{\text{K}m}=m\sqrt{1-\frac{r^2}{l^2}}
$$
Which means that lowering the mass from the center to a position very close to cosmological horizon allows us to convert almost all of rest energy of the mass into useful work. But once again this is not a “free energy”: it is not possible to extract more work than the rest energy of the mass being lowered and if we try to raise back this mass we would need to expend work to do it while the losses that would always be present would ensure that the efficiency of energy conversion is always less than unity.
Qualitatively similar situation would arise in any cosmological model that has cosmological horizons: reservoir of energy available for extraction is always finite and is determined by available matter that we can tether to us before it falls through cosmological horizon.
References
The first paper is not specifically about extraction of energy from universe expansion but a general discussion about physics within long-term cosmological evolution:

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*Dyson, F. J. (1979). Time without end: Physics and biology in an open universe. Reviews of Modern Physics, 51(3), 447, doi:10.1103/RevModPhys.51.447, free pdf.

The rest of the references require familiarity with machinery of general relativity at least at the level of introductory textbooks:

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*Harrison, E. R. (1995). Mining energy in an expanding universe. The Astrophysical Journal, 446, 63. online pdf.


*Davis, T. M., Lineweaver, C. H., & Webb, J. K. (2003). Solutions to the tethered galaxy problem in an expanding universe and the observation of receding blueshifted objects. American Journal of Physics, 71(4), 358-364, doi:10.1119/1.1528916, arXiv:astro-ph/0104349


*MacLaurin, C. (2019). Cosmic cable. arXiv:1911.08726.
And finally lecture notes on the energy in GR:

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*Chrusciel, P. T. (2012). Lectures on energy in general relativity, kraków, march-april 2010, pdf online.

