  We study the asymptotic behavior of the (modified) superconformal index for
4d $\mathcal{N} = 1$ gauge theory. By considering complexified chemical
potential, we find that the `high-temperature limit' of the index can be
written in terms of the conformal anomalies $3c-2a$. We also find macroscopic
entropy from our asymptotic free energy when the Hofman-Maldacena bound $1/2 <
a/c < 3/2$ for the interacting SCFT is satisfied. We study $\mathcal{N} = 1$
theories that are dual to AdS$_5 \times Y^{p, p}$ and find that the Cardy limit
of our index accounts for the Bekenstein-Hawking entropy of large black holes.
