In this manuscript, we evaluate the validity of the horizon first law in the framework of ffalse(R,scriptGfalse)$f(R, \mathcal {G})$ 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 1st law as a result of equations of motion in ffalse(R,scriptGfalse)$f(R,\mathcal {G})$ gravity, will have significant importance in our illustration. The derivation of black hole energies and entropies in the framework of specific ffalse(R,scriptGfalse)$f(R, \mathcal {G})$ 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 ffalse(R,scriptGfalse)$f(R, \mathcal {G})$ gravity. Moreover, we shall consider a flat FLRW model corresponding to ffalse(R,scriptGfalse)$f(R,\mathcal {G})$ 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.