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We consider a spatially flat Friedmann-Lemaitre-Robertson-Walker space timeand investigate the second law and the generalized second law of thermodynamicsfor apparent horizon in generalized modified Gauss Bonnet theory of gravity(whose action contains a general function of Gauss Bonnet invariant and theRicci scalar: $F(R,G)$). By assuming that the apparent horizon is in thermalequilibrium with the matter inside it, conditions which must be satisfied by$F(R,G)$ are derived and elucidated through two examples: a quasi-de Sitterspace-time and a universe with power law expansion.

Cosmological entropy and generalized second law of thermodynamics in $F(R,G)$ theory of gravity