It depends on what exactly you mean with "... non-zero energy", but the best answer is probably "no". On the one hand, in vacuum, an accelerating expanding uni

It depends on what exactly you mean with "... non-zero energy", but the best answer is probably "no". On the one hand, in vacuum, an accelerating expanding universe's geometry breaks time-translation invariance. Let's look at the vacuum geometry with a pair of test photons in the otherwise empty test universe. In the early universe we have the photons widely spacelike-separated but on trajectories where they will ultimately collide in the far future. The photons in the future will be redshifted compared to their wavelengths in the past, with the redshift proportional to the expansion history of the test universe. (In our universe, the cosmic microwave background photons have behaved this way when two CMB photons meet at a detector). Since photon energy can be calculated E = (hc)/\lambda, where \lambda is the photon wavelength, a redshift corresponds to a decrease in the photon energy. Where did this energy go? One can simply say that constancy of photon energy is premised on time-translation invariance, which is not a large-scale feature of an accelerating expanding universe. This position is just that thermodynamics was developed to describe engineering phenomena in the 19th century, and not to describe the motion of clusters of galaxies over the course of billions of years. Since Noether, thermodynamics's first law has its foundations in time-translation invariance, which we only have in some spacetimes (e.g., Minkowski's flat space, the spacetime of special relativity) or in local pieces of general curved spacetime. In the absence of strict time-translation invariance globally, the first law of thermodynamics is just a weak-gravitational-field approximation. Sean Carroll expands upon some of this here : https://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/ https://www.preposterousuniverse.com/blog/2010/02/22/energy-... wherein he also discusses an effectively pseudogravitational-field view of our test spacetime's geometry, in order to think of our pair of converging photons donating energy to that (pseudo-)gravitational field. As he says, such a view, which leans on a notion of gravitational potential energy which is simply absent in General Relativity, is often unhelpful. On the other hand, one can take a different approach and treat \Lambda, the cosmological constant, as a matter field rather than as a geometrical feature of the accelerating expanding spacetime. When we do this, then we are left with either an inertially-expanding spacetime, or with a flat spacetime, and in the stress-energy tensor we have what makes that acceleration or acceleration + expansion field cause everything in the stress-energy tensor (including those fields themselves) to fly apart. It can be reasonable enough to do this. Einstein and Schrödinger discussed this in correspondence during the first world war : https://arxiv.org/abs/1211.6338 https://arxiv.org/abs/1211.6338 wherein they seem to agree that it is a matter of taste. We know from work in the more modern Hamiltonian formulation of General Relativity that we can't do this generally, but there are large families of spacetimes and matter content that are amenable to this approach of encoding aspects of the dynamical geometry as a form matter we can describe with canonical position & momentum. We can also note here that the distribution of matter in a spacetime can pick out preferred foliations or even preferred frames of reference. In our standard cosmology, which you are asking about, we treat the distribution of matter isotropic and homogeneous, where each form of matter is a perfect fluid with a density and a pressure. As the universe expands (with or without acceleration of expansion) the matter and radiation fluids dilute away, their density dropping over time. We encode the acceleration of expansion as an additional fluid which simply does not dilute away: it retains a constant density even as space expands or as the expansion accelerates, and is in effect the energy of the vacuum. "Matter tells spacetime how to curve" means that the matter fields, important when they are dense in the early universe, become less important in the later universe. In the later universe, the additional expansion/acceleration field dominates, causing the spatial distances between inertial observes (we can talk about spatial distances in the preferred frame of reference our isotropic & homogeneous fluids pick out) to widen. In this view, the expansion of the universe is an adiabatic process on the matter and radiation fields, but the additional field -- a vacuum energy -- retains a constant energy-density. We can also consider the pressure of these spacetime-filling fluids. All of the (diluting) matter has a positive pressure. If you create a locally-overdense bubble of positive pressure fluid, it will tend to become denser as it collapses gravitationally. Local overdensities of matter (including dark matter, which also has positive pressure, and also dilutes away with expansion) in the early universe are responsible for structure formation, wherein gasses collapse forming the first stars. The (nondiluting) dark energy field by contrast has a negative pressure. Local overdensities of that field tend to anti-collapse gravitationally, becoming less dense. This is peculiar, but you can think of what you see in boiling water, where the readily observed water vapour voids are less dense than the water. Voids from overdensities of dark energy in the early universe may also be important in structure formation, like underdensities of matter in the simpler model in the couple paragraphs above. This view gets messy when treated formally: you do not need to generate more positive-pressure matter into a collapsing overdensity, it just gathers up into a denser and denser arrangement. (We ignore details of black holes). However, it arguably has a virtue that it at least partially (depends on details of the field dynamics and a time-dependent equation-of-state parameter; for example we may need a decay term where "overdense" dark energy decays into normally-dense dark energy) recovers from the breaking of time-translation symmetry on a cosmological scale. Some of the mess is to keep a "fifth" fundamental force from appearing and causing long-range effects which we simply do not observe. Carroll (again) discusses some of this here : https://www.preposterousuniverse.com/blog/2013/11/16/why-does-dark-energy-make-the-universe-accelerate/ https://www.preposterousuniverse.com/blog/2013/11/16/why-doe... Forgive any errors in sign and unclarities. My system has decided to become uncooperative during the editing pass and I'm just hitting reply so the text above doesn't vanish.