  The counting of BPS states in four-dimensional ${\cal N}=1$ theories has
attracted a lot of attention in recent years. For superconformal theories,
these states are in one-to-one correspondence with local operators in various
short representations. The generating function for this counting problem has
branch cuts and hence several Cardy-like limits, which are analogous to
high-temperature limits. Particularly interesting is the second sheet, which
has been shown to capture the microstates and phases of supersymmetric black
holes in AdS$_5$. Here we present a 3d Effective Field Theory (EFT) approach to
the high-temperature limit on the second sheet. We use the EFT to derive the
behavior of the index at orders $\beta^{-2},\beta^{-1},\beta^0$. We also make a
conjecture for $O(\beta)$, where we argue that the expansion truncates up to
exponentially small corrections. An important point is the existence of vector
multiplet zero modes, unaccompanied by massless matter fields. The runaway of
Affleck-Harvey-Witten is however avoided by a non-perturbative confinement
mechanism. This confinement mechanism guarantees that our results are robust.
