  Perturbation theory plays an essential role in the realm of physics,
especially when it is difficult or impossible to obtain exact solutions.
Incorporating higher derivative terms into the gravitational action leads to an
important perturbation problem. In this paper, we demonstrate that the first
order corrections to black hole thermodynamics, caused by any higher derivative
terms, can be achieved without explicitly solving the modified metric. This
enhances a simple approach discovered two decades ago and resolves the
controversy surrounding it. Astonishingly, we observe that the perturbation
theory of quantum mechanics also exhibits a similar behavior, which reveals the
existence of a universal theoretical structure underlying the perturbation
theories of black hole thermodynamics, quantum mechanics and even other
disciplines.
