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The thermodynamics of the Universe is restudied by requiring itscompatibility with the holographic-style gravitational equations which governthe dynamics of both the cosmological apparent horizon and the entire Universe,and possible solutions are proposed to the existent confusions regarding theapparent-horizon temperature and the cosmic entropy evolution. We start fromthe generic Lambda Cold Dark Matter ($\Lambda$CDM) cosmology of generalrelativity (GR) to establish a framework for the gravitational thermodynamics.The Cai-Kim Clausius equation for the isochoric process of an instantaneousapparent horizon indicates that, the Universe and its horizon entropies encodethe \emph{positive heat out} thermodynamic sign convention, which encourages usto adjust the traditional positive-heat-in Gibbs equation into thepositive-heat-out version $dE_m=-T_mdS_m-P_mdV$. It turns out that the standardand the generalized second laws (GSLs) of nondecreasing entropies are alwaysrespected by the event-horizon system as long as the expanding Universe isdominated by nonexotic matter $-1\leq w_m\leq 1$, while for theapparent-horizon simple open system the two second laws hold if $-1\leqw_m<-1/3$; also, the artificial local equilibrium assumption is abandoned inthe GSL. All constraints regarding entropy evolution are expressed by theequation of state parameter, which show that from a thermodynamic perspectivethe phantom dark energy is less favored than the cosmological constant and thequintessence. Finally, the whole framework is extended from GR and $\Lambda$CDMto modified gravities with field equations $R_{\mu\nu}-Rg_{\mu\nu}/2=8\piG_{\text{eff}} T_{\mu\nu}^{\text{(eff)}}$. Furthermore, this paper argues thatthe Cai-Kim temperature is more suitable than Hayward, and theBekenstein-Hawking and Wald entropies cannot unconditionally apply to the eventand particle horizons.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d, particles, fields, gravitation, and cosmology

Apparent horizon and gravitational thermodynamics of the Universe:  Solutions to the temperature and entropy confusions, and extensions to  modified gravity