  In modified theories of gravity, higher curvature terms may be added to the
Einstein-Hilbert action. Conventionally, the effects of the higher curvature
terms on the black hole thermodynamics are rather difficult to obtain. In this
paper, we show that, at least at the first order level, the corrections to the
thermodynamics of the Schwarzschild black hole can be easily obtained, without
solving the modified black hole metric. We examine some specific examples,
which produces the known results in the literature, as well as those previously
unknown. Furthermore, we find some nonperturbative exact results by
generalizing the first-order analysis. In particular, a novel relation between
the Euclidean integrals of the Einstein--Hilbert term and the higher curvature
terms is found and proved.
