  Both classical and quantum arguments suggest that if Barrow entropy is
correct, its index $\delta$ must be energy dependent, which would affect the
very early universe. Based on thermodynamic stability that sufficiently large
black holes should not fragment, we argue that Barrow entropy correction must
be small except possibly at the Planckian regime. Furthermore, the fact that a
solar mass black hole does not fragment implies an upper bound $\delta \lesssim
O(10^{-3})$, which surprisingly lies in the same range as the bound obtained
from some cosmological considerations assuming fixed $\delta$. This indicates
that allowing $\delta$ to run does not raise its allowed value. We briefly
comment on the case of Kaniadakis entropy.
