The Thermodynamics of the Gravity from Entropy Theory: from the Hamiltonian to applications in Cosmology

Abstract: The Gravity from Entropy (GfE) action posits that the fundamental nature of gravity is information encoded in the metric degrees of freedom. This statistical mechanics theory leads to modified gravity equations that reduce to the Einstein equations in the limit of low energy and small curvature. Here we embrace a thermodynamic point of view to derive the Hamiltonian associated with this theory. Focusing on isotropic spacetimes, we derive the thermodynamic properties of the GfE theory. We reveal that the FRW metrics are associated with $k$-temperature and $k$-pressure which are related to their local Geometric Quantum Relative Entropy (GQRE) and their local energy by the first law of GfE thermodynamics. The thermodynamics of the GfE theory is illustrated for the case of Friedmann universes that are solutions of the GfE equations of motion in the low energy, small curvature limit. We show that while the total GQRE for unit volume is not increasing, coherently with its relative entropy nature, the total entropy of Friedmann universes is not decreasing in time.