Thermodynamic Behavior of Friedmann Equation at Apparent Horizon of FRW Universe

TL;DR

The paper demonstrates that the differential form of the Friedmann equations at the apparent horizon of an FRW universe can be cast into a universal thermodynamic identity $dE = TdS + WdV$, where $E$ is the enclosed energy, $S$ is the horizon entropy, $T$ is the horizon temperature, and $W=(\rho-P)/2$ is the work density. This structure holds not only in Einstein gravity, but also extends to Gauss-Bonnet and Lovelock gravities, with entropy expressions consistent with black hole thermodynamics in each theory and with $E = \rho V$ as the energy term in the first law (distinct from Misner-Sharp energy in higher-curvature gravities). The results reinforce the deep link between gravitational dynamics and horizon thermodynamics, supporting a holographic or emergent view of gravity, and illuminate how higher-curvature corrections modify horizon entropy while preserving the thermodynamic form of the field equations at the apparent horizon.

Abstract

It is shown that the differential form of Friedmann equation of a FRW universe can be rewritten as a universal form $dE = TdS + WdV$ at apparent horizon, where $E$ and $V$ are the matter energy and volume inside the apparent horizon (the energy $E$ is the same as the Misner-Sharp energy in the case of Einstein general relativity), $W=(ρ-P)/2$ is the work density and $ρ$ and $P$ are energy density and pressure of the matter in the universe, respectively. From the thermodynamic identity one can derive that the apparent horizon has associated entropy $S= A/4G$ and temperature $T = κ/ 2π$ in Einstein general relativity, where $A$ is the area of apparent horizon and $κ$ is the surface gravity at apparent horizon. We extend our procedure to the Gauss-Bonnet gravity and more general Lovelock gravity and show that the differential form of Friedmann equations in these gravities can also be rewritten to thee universal form $dE = TdS + WdV$ at the apparent horizon with entropy $S$ being given by expression previously known via black hole thermodynamics.