Effective Temperature of the FRW Universe
Shi-Bei Kong
TL;DR
The paper defines an effective temperature Teff for the FRW universe via Teff = dE/dS to resolve sign ambiguities in the first-law form and to pair with entropy S. It computes Teff and the corresponding free energy for N-dimensional FRW universes in Einstein, Gauss-Bonnet, and Lovelock gravity. In four dimensions Teff reduces to 1/(4π RA), matching the Hawking temperature of Schwarzschild black holes, while in higher dimensions Teff acquires gravity-theory dependent corrections; Gauss-Bonnet adds a 1/RA^3 term and Lovelock introduces higher-order contributions with rich behavior for certain couplings; explicit N=7 and N=8 cases are given. The results provide a thermodynamic lens on cosmological horizons in modified gravity and hint at extensions to dynamical horizons and Wald entropy comparisons.
Abstract
In this paper, we give a new definition of temperature for the FRW(Friedmann-Robertson-Walker) universe, i.e. the effective temperature $T_{eff}:=dE/dS$, where $E$ is the energy and $S$ is the entropy of the FRW universe. Based on this definition, we get the effective temperature for the $N$-dimensional FRW universe in Einstein gravity, Gauss-Bonnet gravity and Lovelock gravity. We find that for the 4-dimensional FRW universe, the effective temperature is always $T_{eff}=1/(4πR_A)$, which is exactly the same form with the Hawking temperature of the Schwarzschild black hole. In higher-dimensional FRW universe, the form of the effective temperature depends on the choices of the gravitational theories or the corresponding coupling constants. We also get the free energy of the FRW universe in the three theories of gravity.