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In modified theories of gravity, higher curvature terms may be added to theEinstein-Hilbert action. Conventionally, the effects of the higher curvatureterms on the black hole thermodynamics are rather difficult to obtain. In thispaper, we show that, at least at the first order level, the corrections to thethermodynamics of the Schwarzschild black hole can be easily obtained, withoutsolving the modified black hole metric. We examine some specific examples,which produces the known results in the literature, as well as those previouslyunknown. Furthermore, we find some nonperturbative exact results bygeneralizing the first-order analysis. In particular, a novel relation betweenthe Euclidean integrals of the Einstein--Hilbert term and the higher curvatureterms is found and proved.

On the first order corrections to the black hole thermodynamics in higher curvature theories of gravity