  We study the supersymmetric partition function of 4d supersymmetric gauge
theories with a U(1) R-symmetry on Euclidean $S^3\times S_\beta^1$, with $S^3$
the unit-radius squashed three-sphere, and $\beta$ the circumference of the
circle. For superconformal theories, this partition function coincides (up to a
Casimir energy factor) with the 4d superconformal index.
  The partition function can be computed exactly using supersymmetric
localization of the gauge theory path-integral. It takes the form of an
elliptic hypergeometric integral, which may be viewed as a matrix-integral over
the moduli space of the holonomies of the gauge fields around $S_\beta^1$. At
high temperatures ($\beta\to 0$, corresponding to the hyperbolic limit of the
elliptic hypergeometric integral) we obtain from the matrix-integral a quantum
effective potential for the holonomies. The effective potential is proportional
to the temperature. Therefore the high-temperature limit further localizes the
matrix-integral to the locus of the minima of the potential. If the effective
potential is positive semi-definite, the leading high-temperature asymptotics
of the partition function is given by the formula of Di Pietro and Komargodski,
and the subleading asymptotics is connected to the Coulomb branch dynamics on
$R^3\times S^1$. In theories where the effective potential is not positive
semi-definite, the Di Pietro-Komargodski formula needs to be modified. In
particular, this modification occurs in the SU(2) theory of
Intriligator-Seiberg-Shenker, and the SO(N) theory of Brodie-Cho-Intriligator,
both believed to exhibit "misleading" anomaly matchings, and both believed to
yield interacting superconformal field theories with $c<a$.
  Two new simple tests for dualities between 4d supersymmetric gauge theories
emerge as byproducts of our analysis.
