The thermodynamics of the Universe is restudied by requiring its compatibility with the holographic-style gravitational equations which 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 (ΛCDM) cosmology of general relativity (GR) to establish a framework for the gravitational thermodynamics. The Cai-Kim Clausius equation δQ = T A dS A = −dE A = −A A ψ 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 show 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 GR and ΛCDM to modified gravities with field equations R µν − Rg µν /2 = 8πG eff T (eff) µν . 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.

Inspired by the Wald-Kodama entropy S = A/(4G eff ) where A is the horizon area and G eff is the effective gravitational coupling strength in modified gravity with field equationµν , we develop a unified and compact formulation in which the Friedmann equations can be derived from thermodynamics of the Universe. The Hawking and Misner-Sharp masses are generalized by replacing Newton's constant G with G eff , and the unified first law of equilibrium thermodynamics is supplemented by a nonequilibrium energy dissipation term E which arises from the revised continuity equation of the perfect-fluid effective matter content and is related to the evolution of G eff . By identifying the mass as the total internal energy, the unified first law for the interior and its smooth transit to the apparent horizon yield both Friedmann equations, while the nonequilibrium Clausius relation with entropy production for an isochoric process provides an alternative derivation on the horizon. We also analyze the equilibrium situation G eff = G = constant, provide a viability test of the generalized geometric masses, and discuss the continuity/conservation equation. Finally, the general formulation is applied to the FRW cosmology of minimally coupled f (R), generalized Brans-Dicke, scalar-tensor-chameleon, quadratic, f (R, G) generalized Gauss-Bonnet and dynamical Chern-Simons gravity. In these theories we also analyze the f (R)-Brans-Dicke equivalence, find that the chameleon effect causes extra energy dissipation and entropy production, geometrically reconstruct the mass ρ m V for the physical matter content, and show the self-inconsistency of f (R, G) gravity in problems involving G eff .

For a large class of scalar-tensor-like gravity whose action contains nonminimal couplings between a scalar field φ(x α ) and generic curvature invariants {R} beyond the Ricci scalar R = R α α , we prove the covariant invariance of its field equation and confirm/prove the local energy-momentum conservation. These φ(x α ) − R coupling terms break the symmetry of diffeomorphism invariance under a particle transformation, which implies that the solutions to the field equation should satisfy the consistency condition R ≡ 0 when φ(x α ) is nondynamical and massless. Following this fact and based on the accelerated expansion of the observable Universe, we propose a primary test to check the viability of the modified gravity to be an effective dark energy, and a simplest example passing the test is the "Weyl/conformal dark energy".

According to Lovelock's theorem, the Hilbert-Einstein and the Lovelock actions are indistinguishable from their field equations. However, they have different scalar-tensor counterparts, which correspond to the Brans-Dicke and the Lovelock-Brans-Dicke (LBD) gravities, respectively. In this paper the LBD model of alternative gravity with the Lagrangian densitywhere * RR and G respectively denote the topological Chern-Pontryagin and Gauss-Bonnet invariants. The field equation, the kinematical and dynamical wave equations, and the constraint from energy-momentum conservation are all derived. It is shown that, the LBD gravity reduces to general relativity in the limit ω L → ∞ unless the "topological balance condition" holds, and in vacuum it can be conformally transformed into the dynamical Chern-Simons gravity and the generalized Gauss-Bonnet dark energy with Horndeski-like or Galileon-like kinetics. Moreover, the LBD gravity allows for the late-time cosmic acceleration without dark energy. Finally, the LBD gravity is generalized into the Lovelock-scalar-tensor gravity, and its equivalence to fourth-order modified gravities is established. It is also emphasized that the standard expressions for the contributions of generalized Gauss-Bonnet dependence can be further simplified.

Lessons from f (R, R Faculty of Science, Memorial University, St. John's, Newfoundland, Canada, A1C 5S7 Ivan Booth † Department of Mathematics and Statistics, Memorial University, St. John's, Newfoundland, Canada, A1C 5S7 This paper studies a generic fourth-order theory of gravity with Lagrangian density Gauss-Bonnet gravity with G denoting the Gauss-Bonnet invariant. We use Noether's conservation law to study the f (R 1 , R 2 . . . , R n , L m ) model with nonminimal coupling between L m and Riemannian invariants R i , and conjecture that the gradient of nonminimal gravitational coupling strength ∇ µ f L m is the only source for energy-momentum nonconservation.This conjecture is applied to the f (R, R Focusing on modifications of GR, the original Lagrangian density can be modified in two ways: (1) extending its dependence on the curvature invariants, and (2) In all these models, the spacetime geometry remains minimally coupled to the matter Lagrangian density L m .On the other hand, following the spirit of nonminimal f (R)L d coupling in scalar-field dark-energy models [9], for modified theories of gravity an extra term λf (R)L m was respectively added to the standard actions of GR and f (R) + 2κL m gravity in [10] and [11], which represents nonminimal curvature-matter coupling between R *