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

The thermodynamics of the Universe is restudied by requiring its compatibility with the holographic-style gravitational equations that govern the dynamics of both the cosmological apparent horizon and the entire Universe, and possible solutions are proposed to the existent confusions regarding the apparent-horizon temperature and the cosmic entropy evolution. We start from the generic Lambda cold dark matter cosmology of general relativity to establish a framework for the gravitational thermodynamics. The Cai-Kim Clausius equation delta Q = T(A)dS(A) = -dE(A) = -A(A)Psi(t) for the isochoric process of an instantaneous apparent horizon indicates that the Universe and its horizon entropies encode the positive-heat-out thermodynamic sign convention, which encourages us to adjust the traditional positive-heat-in Gibbs equation into the positive-heat-out version dE(m) = -T(m)dS(m) - P(m)dV. It turns out that the standard and the generalized second laws (GSLs) of nondecreasing entropies are always respected by the event-horizon system as long as the expanding Universe is dominated by nonexotic matter -1 <= w(m) <= 1, while for the apparent-horizon simple open system, the two second laws hold if -1 <= w(m) < -1/3; also, the artificial local equilibrium assumption is abandoned in the GSL. All constraints regarding entropy evolution are expressed by the equation-of-state parameter, which shows that from a thermodynamic perspective the phantom dark energy is less favored than the cosmological constant and the quintessence. Finally, the whole framework is extended from general relativity and Lambda cold dark matter to modified gravities with field equations R-mu nu - Rg(mu nu)/2 = 8 pi G(eff)T(mu nu)((eff)). Furthermore, this paper argues that the Cai-Kim temperature is more suitable than Hayward, both temperatures are independent of the inner or outer trappedness of the apparent horizon, and the Bekenstein-Hawking and Wald entropies cannot unconditionally apply to the event and particle horizons.