  In this paper we discuss the supersymmetric localization of the 4D
$\mathcal{N}=2$ off-shell gauged supergravity in the background of the
$\textrm{AdS}_4$ neutral topological black hole, which is the gravity dual of
the ABJM theory defined on the boundary $\textrm{S}^1 \times \mathbb{H}^2$. We
compute the large-$N$ expansion of the supergravity partition function. The
result gives the black hole entropy with the logarithmic correction, which
matches the previous result of the entanglement entropy of the ABJM theory up
to some stringy effects. Our result is consistent with the previous on-shell
one-loop computation of the logarithmic correction to black hole entropy. It
provides an explicit example of the identification of the entanglement entropy
of the boundary conformal field theory with the bulk black hole entropy beyond
the leading order given by the classical Bekenstein-Hawking formula, which
consequently tests the AdS/CFT correspondence at the subleading order.
