Horizon Thermodynamics in f(R,G)$f(R,\mathcal {G})$ Gravity

In this manuscript, we evaluate the validity of the horizon first law in the framework of f(R,G)$f(R, \mathcal {G})$ gravity. We propose that the energy and entropy of a black hole must be consistent with their relating integration values. The feasibility of the newly proposed horizon 1(st) law as a result of equations of motion in f(R,G)$f(R,\mathcal {G})$ gravity, will have significant importance in our illustration. The derivation of black hole energies and entropies in the framework of specific f(R,G)$f(R, \mathcal {G})$ models will be helpful in future research. We study that how the positivity of energy and entropy can impose additional restrictions on the domain of f(R,G)$f(R, \mathcal {G})$ gravity. Moreover, we shall consider a flat FLRW model corresponding to f(R,G)$f(R,\mathcal {G})$ theory and will derive the related Friedman equations. The Cai-Kim technique will then be used to evaluate the expression for energy flux passing through the horizon in insignificantly small time. In addition, we use the Clausius relation related to the evident boundary as well as the Cai-Kim technique and the relating Friedmann equations, to find the expression for horizon entropy.