A 4d N=1 Cardy Formula

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 $1/2 < a/c < 3/2$fortheinteractingSCFTissatisfied\mathrm{.}Westudy$$\mathcal{N}$ = 1$theoriesthataredualtoAdS$_5 \times Y^{p, p}$ and find that the Cardy limit of our index accounts for the Bekenstein-Hawking entropy of large black holes.