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$f(R)$-Modified Gravity, Wald Entropy, and the Generalized Uncertainty Principle

Wald's entropy formula allows one to find the entropy of black holes' event horizon within any diffeomorphism invariant theory of gravity. When applied to general relativity, the formula yields the Bekenstein-Hawking result but, for any other gravitational action that departs from the Hilbert action, the resulting entropy acquires an additional multiplicative factor that depends on the global geometry of the background spacetime. On the other hand, the generalized uncertainty principle (GUP) has extensively been recently used to investigate corrections to the Bekenstein-Hawking entropy formula, with the conclusion that the latter always comes multiplied by a factor that depends on the area of the event horizon. We show, by considering the case of an $f(R)$-modified gravity, that the usual black hole entropy derivation based on the GUP might be modified in such a way that the two methods yield the same corrections to Bekenstein-Hawking formula. The procedure turns out to be an interesting method for seeking modified gravity theories. Two different versions of the GUP are used and it is found that only one of them yields a viable modified gravity model. Conversely, it is possible to find a general formulation of the GUP that would reproduce Wald entropy formula for any $f(R)$-theory of gravity.