The far future is surprisingly simple and yet very interesting from a scientific perspective. In due time, all the matter and radiation of the universe will be absorbed by the cosmological horizon, which grows in response. Nothing will be left other than slowly evaporating supermassive black holes. Black holes are believed to be the densest and most lasting objects in the universe. We study the thermodynamics of our local universe in this future. The local group is expected to collapse to a supermassive black hole that will slowly evaporate in an empty universe dominated by a cosmological constant. In our paper we look at the dynamics between the extremely cold cosmological horizon and the hotter black hole horizon. We discuss how heat and entropy produced by black hole are absorbed by the cosmological horizon and find that even though the presence of the black hole initially depressed the total entropy, the total entropy increases as the black hole evaporates and, of course, heat flows from the hotter black hole horizon to the colder cosmological horizon.
So, what is the end state of the universe? If the cosmological constant is a true constant, then the universe will reach its final thermodynamics equilibrium state in empty deSitter, i.e., empty space that expands forever. However, if a slow rolling scalar field mimics the cosmological constant, this scalar field could eventually reach the bottom of its potential and then the universe would stop expanding becoming flat. Stephen Hawking states that since flat space has no horizon, it also has zero entropy. We argue that it has divergent entropy because it could be indistinguishable from a space with a very small cosmological constant; the entropy of deSitter depends on the inverse of the cosmological constant. With this assumption, flat space would have a infinitely large entropy, which would be consistent with the second law of thermodynamics; if flat space could be the final state of our universe at the end of the slow roll period, it should have higher entropy than deSitter.
We also discuss the Weyl curvature hypothesis of Roger Penrose who states that the difference in entropy between the initial and final state of the universe is related to the growth of the Weyl curvature. The Weyl curvature is small at the beginning of the universe, but grows with the production of singularities. However, empty deSitter has zero Weyl curvature. Empty deSitter is the thermodynamics equilibrium (final state) of our universe for a true cosmological constant. We modify the conjecture by stating that the entropy does not come directly from the Weyl tensor, but from a coarse-graining over the states of the Weyl tensor. The cosmological constant limits the wavelength of the gravitational wave modes. When a black hole evaporates, the cosmological horizon grows allowing more gravitational wave modes. This increases the number of available microstates in the Weyl tensor and hence the entropy of the space. We emphasize that the cosmological constant is not just an energy scale, but an entropy scale as well. It plays a very important role in black hole thermodynamics that still has to be understood.