  We study the superconfomal index of four-dimensional toric quiver gauge
theories using a Bethe Ansatz approach recently applied by Benini and Milan.
Relying on a particular set of solutions to the corresponding Bethe Ansatz
equations we evaluate the superconformal index in the large $N$ limit, thus
avoiding to take any Cardy-like limit. We present explicit results for theories
arising as a stack of $N$ D3 branes at the tip of toric Calabi-Yau cones: the
conifold theory, the suspended pinch point gauge theory, the first del Pezzo
theory and $Y^{p,q}$ quiver gauge theories. For a suitable choice of the
chemical potentials of the theory we find agreement with predictions made for
the same theories in the Cardy-like limit. However, for other regions of the
domain of chemical potentials the superconformal index is modified and
consequently the associated black hole entropy receives corrections. We work
out explicitly the simple case of the conifold theory.
