  The entropy of a Schwarzschild black hole, as computed via the semiclassical
Euclidean path integral in a stationary phase approximation, is determined not
by the on-shell value of the action (which vanishes), but by the
Gibbons--Hawking--York boundary term evaluated on a suitable hypersurface,
which can be chosen arbitrarily far away from the horizon. For this reason, the
black hole singularity seemingly has no influence on the Bekenstein--Hawking
area law. In this Essay we estimate how a regular black hole core, deep inside
a Euclidean black hole of mass $M$ and generated via a UV regulator length
scale $\ell > 0$, affects the black hole entropy. The contributions are
suppressed by factors of $\ell/(2GM)$; demanding exact agreement with the area
law as well as a self-consistent first law of black hole thermodynamics at all
orders, however, demands that these contributions vanish identically via
uniformly bounded curvature. This links the limiting curvature hypothesis to
black hole thermodynamics.
