  We study a limit of the superconformal index of the ABJM theory on $S^1\times
S^2$ in which the size of the circle is much smaller than the radius of the
two-sphere. We derive closed form expressions for the two leading terms in this
Cardy-like limit which are valid to all orders in the $1/N$ expansion. These
results are facilitated by a judicious rewriting of the superconformal index
which establishes a connection with the Bethe Ansatz Equations that control the
topologically twisted index. Using the same technique we extend these results
to the superconformal index of another holographic theory: 3d $\mathcal{N}=4$
SYM coupled to one adjoint and $N_f$ fundamental hypermultiplets. We discuss
the implications of our results for holography and the physics of charged
rotating black holes in AdS$_4$.
