Universality of entropy principle for a general diffeomorphism-covariant purely gravitational theory

Thermodynamics plays an important role in gravitational theories. It is a principle independent of the dynamics, and there still no rigorous proof to show that it consistent with dynamical principle. We consider self-gravitating perfect fluid system general diffeomorphism-covariant purely theory. Based on Noether charge method proposed by Iyer Wald, considering static off/on-shell variational configurations which satisfy constraint equation, we rigorously prove extrema total entropy inside compact region for fixed particle number demands configuration on-shell solution after introduce some appropriate boundary conditions, i.e., also satisfies spatial equations. This means stores same information as equation configuration. Our universal holds any theories, such Einstein gravity, f(R) Lovelock f(Gauss-Bonnet) gravity Einstein-Weyl gravity. result shows consistency between ordinary thermodynamics dynamics.