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