How barrow entropy modifies gravity: with comments on Tsallis entropy

Barrow proposed that the area law of the horizon entropy might receive a "fractal correction" S $\propto$ A${}^{1+\Delta /2}$ due to quantum gravitational effects, with 0 $⩽$ Δ $⩽$ 1 measures the deviation from the standard area law. While such a modification has been widely studied in the literature, its corresponding theory of gravity has not been discussed. We follow Jacobson's approach to derive the modified gravity theory (interpreted as an effective theory), and find that in the stationary case the resulting theory only differs from general relativity by a re-scaled cosmological constant. Consequently in asymptotically flat stationary spacetimes the theory is identical to general relativity. The theory is not applicable when there is no horizon; the multi-horizon case is complicated. We emphasize on the importance of identifying the correct thermodynamic mass in a theory with modified thermodynamics to avoid inconsistencies. We also comment on the Hawking evaporation rate beyond the effective theory. In addition, we show that the Bekenstein bound is satisfied if the thermodynamic mass is used as the energy, up to a constant prefactor. We briefly comment on the Tsallis entropy case as well. Interestingly, for the latter, the requirement that Bekenstein bound holds imposes a lower bound on the non-extensive parameter: δ > 1/2, which unfortunately rules out the previously suggested possibility that the expansion of the universe can accelerate with normal matter field alone.