Do Black Holes With Generalized Entropy Violate Bekenstein Bound?

In general yes, but also not quite. It is known that if the Bekenstein-Hawking entropy is replaced by some kind of generalized entropy, then the Bekenstein bound may be grossly violated. In this work, we show that this undesired violation can be avoided if we employ the equivalence between generalized entropy and varying-$G$ gravity (GEVAG). In this approach, modifying entropy necessarily also modifies gravity (as one should expect if gravity is indeed inherently tied to thermodynamics), which leads to an effective gravitational "constant" $G_\text{eff}$ that is area-dependent, and a thermodynamic energy that is distinct from the ADM mass. We show that a relaxed Bekenstein bound of the form $S \leqslant CRE$ is always satisfied, albeit the coefficient $C$ is no longer $2\pi$.