  Recently it was discovered that twisted superconformal index ${\mathcal{I}}$
can be used to understand the Bekenstein-Hawking entropy of magnetically
charged black holes in AdS spacetime. In this paper we apply the so-called
$\mathcal{I}$-extremization procedure to three-dimensional gauge field theories
and their geometric dual, focusing in particular on the seven-dimensional
Sasaki-Einstein manifold $M^{1,1,1}$. We generalize recent studies on relations
among toric geometry, variational principles, and black hole entropy to the
case of AdS$_2 \times Y_9$, where $Y_9$ is a fibration of toric Sasaki-Einstein
manifold $M^{1,1,1}$ over a two-dimensional Riemann surface $\Sigma_g$. The
nine-dimensional variational problem is given in terms of an entropy
functional. In order to illustrate the computations as explicitly as possible,
we consider cases where either only mesonic or baryonic fluxes are turned on.
By employing the operator counting method, we calculate the $S^3$ free energy
and the topologically twisted index $\mathcal{I}$ at large-$N$. The result for
$\mathcal{I}$, it turns out, can be also obtained from the variational
principle of the entropy functional with mesonic fluxes. We also study
asymptotically AdS${}_4$ black holes which are magnetically charged with
respect to the vector field in the Betti multiplet. By extremizing the entropy
functional with baryonic flux, we compute the entropy and find that it agrees
with the entropy of an explicit solution in a four-dimensional gauged
supergravity which is a consistent truncation of eleven-dimensional
supergravity in AdS${}_4\times M^{1,1,1}$.
