Universal relation among Euclidean integrals for black holes in higher-derivative gravity theories

In this paper, we establish a universal equality governing Euclidean integrals of gravitational actions in higher-derivative theories. This relation is shown to hold universally for asymptotically flat black holes in pure gravity, and is generalized to asymptotically anti-de Sitter spacetimes through appropriate regularization. We further examine its validity in systems with matter-gravity coupling, identifying that violations occur only when matter fields exhibit pathological behaviors: divergence at the horizon or nondecaying profiles at infinity. These findings reveal fundamental constraints on gravitational thermodynamics and provide diagnostic tools for identifying ill-behaved matter configurations.