  We compute the on-shell action of static, BPS black holes in AdS$_4$ from
${\cal N}=2$ gauged supergravity coupled to vector multiplets and show that it
is equal to minus the entropy of the black hole. Holographic renormalization is
used to demonstrate that with appropriate boundary conditions on the scalar
fields, the divergent and finite contributions from the asymptotic boundary
vanish. The entropy arises from the extrinsic curvature on $\Sigma_g\times S^1$
evaluated at the horizon, where $\Sigma_g$ may have any genus $g\geq 0$. This
provides a clarification of the equivalence between the partition function of
the twisted ABJM theory on $\Sigma_g\times S^1$ and the entropy of the dual
black hole solutions. It also demonstrates that the complete entropy resides on
the AdS$_2\times \Sigma_g$ horizon geometry, implying the absence of hair for
these gravity solutions.
