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 first determine the entropy of black hole via Wald method, then we derive the energy by using the new horizon first law, the degenerate Legendre transformation, and the gravitational field equations. For application, we consider the quadratic-curvature gravity and firstly calculate the entropy and the energy for a static spherically symmetric black hole, which reduces to the results obtained in literatures for a Schwarzschild-(A)dS black hole.

中文翻译：

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$f(R,R^{\mu\nu}R_{\mu\nu})$理论中的视界热力学

我们研究了新视界第一定律在 $f(R,R^{\mu\nu}R_{\mu\nu})$ 理论中是否仍然成立。对于这个复杂的理论，我们首先通过瓦尔德方法确定黑洞的熵，然后利用新视界第一定律、退化勒让德变换和引力场方程推导出能量。对于应用，我们考虑二次曲率引力，首先计算静态球对称黑洞的熵和能量，这简化为文献中关于 Schwarzschild-(A)dS 黑洞的结果。