Does the validity of the first law of thermodynamics imply that the generalized second law of thermodynamics of the universe is bounded by the event horizon?

In this work we examine the validity of the generalized second law of thermodynamics of the universe with the event horizon as the boundary assuming the first law of thermodynamics. We consider a homogeneous and isotropic model of the universe, filled with perfect fluid having an arbitrary equation of state. We study the validity of the generalized second law both in Einstein and Einstein-Gauss-Bonnet (EGB) gravity.