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Thermodynamics of the apparent horizon in massive cosmology

Applying Clausius relation with energy-supply defined by the unified first law of thermodynamics formalism to the apparent horizon of a massive cosmological model proposed lately, the corrected entropic formula of the apparent horizon is obtained with the help of the modified Friedmann equations. This entropy-area relation, together with the identified internal energy, verifies the first law of thermodynamics for the apparent horizon with a volume change term for consistency. On the other hand, by means of the corrected entropy-area formula and the Clausius relation $δQ=T dS$, the modified Friedmann equations governing the dynamical evolution of the universe are reproduced with the known energy density and pressure of massive graviton. The integration constant is found to correspond to a cosmological term which could be absorbed into the energy density of matter. Having established the correspondence of massive cosmology with the unified first law of thermodynamics on the apparent horizon, the validity of the generalized second law of thermodynamics is also discussed by assuming the thermal equilibrium between the apparent horizon and the matter field bounded by the apparent horizon. It is found that, in the limit $H_c\rightarrow 0$ which recovers the Minkowski reference metric solution in the flat case, the generalized second law of thermodynamics holds if $α_3+4α_4<0$. Apart from that, even for the simplest model of dRGT massive cosmology with $α_3=α_4=0$, the generalized second law of thermodynamics could be violated.