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We investigate whether the new horizon first law still holds in$f(R,R^{\mu\nu}R_{\mu\nu})$ theory. For this complicated theory, we firstdetermine the entropy of black hole via Wald method, then we derive the energyby using the new horizon first law, the degenerate Legendre transformation, andthe gravitational field equations. For application, we consider thequadratic-curvature gravity and firstly calculate the entropy and the energyfor a static spherically symmetric black hole, which reduces to the resultsobtained in literatures for a Schwarzschild-(A)dS black hole.

Horizon thermodynamics in $f(R,R^{\mu\nu}R_{\mu\nu})$ theory