  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 $0\leqslant \Delta \leqslant 1$ 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: $\delta >
1/2$, which unfortunately rules out the previously suggested possibility that
the expansion of the universe can accelerate with normal matter field alone.
