Unified first law of thermodynamics in Gauss-Bonnet gravity on an FLRW background

Employing the thermodynamic unified first law through the thermodynamic-gravity conjecture, in this article, we derive for a FLRW universe the Friedmann equations in the framework of Gauss-Bonnet gravity theory. To do this, we project this generalized first law along the Kodama vector field and along the direction of an orthogonal vector to the Kodama vector. The second Friedmann equation is obtained by projecting on the Kodama vector, while the first is obtained by projecting along the flux on the Cauchy hypersurfaces. This result does not assume a priory temperature and an entropy, so the Clausius relation is not used here. Nevertheless, it is used to obtain the corresponding Gauss-Bonnet entropy. In this way, the validity of the generalized second law of thermodynamics is proved for the Gauss-Bonnet gravity theory.