The thermodynamic evolution of the cosmological event horizon

By manipulating the integral expression for the proper radius $R_e$ of the cosmological event horizon (CEH) in a Friedmann-Robertson-Walker (FRW) universe, we obtain an analytical expression for the change $\dd R_e$ in response to a uniform fluctuation $\dd\rho$ in the average cosmic background density $\rho$. We stipulate that the fluctuation arises within a vanishing interval of proper time, during which the CEH is approximately stationary, and evolves subsequently such that $\dd\rho/\rho$ is constant. The respective variations $2\pi R_e \dd R_e$ and $\dd E_e$ in the horizon entropy $S_e$ and enclosed energy $E_e$ should be therefore related through the cosmological Clausius relation. In that manner we find that the temperature $T_e$ of the CEH at an arbitrary time in a flat FRW universe is $E_e/S_e$, which recovers asymptotically the usual static de Sitter temperature. Furthermore, it is proven that during radiation-dominance and in late times the CEH conforms to the fully dynamical First Law $T_e \drv S_e = P\drv V_e - \drv E_e$, where $V_e$ is the enclosed volume and $P$ is the average cosmic pressure.