  We study 4d $\mathcal{N}=1$ supersymmetric theories on a compact Euclidean
manifold of the form $S^1 \times\mathcal{M}_3$. Partition functions of gauge
theories on this background can be computed using localization, and explicit
formulas have been derived for different choices of the compact manifold
$\mathcal{M}_3$. Taking the limit of shrinking $S^1$, we present a general
formula for the limit of the localization integrand, derived by simple
effective theory considerations, generalizing the result of arXiv:1512.03376.
The limit is given in terms of an effective potential for the holonomies around
the $S^1$, whose minima determine the asymptotic behavior of the partition
function. If the potential is minimized in the origin, where it vanishes, the
partition function has a Cardy-like behavior fixed by $\mathrm{Tr}(R)$, while a
nontrivial minimum gives a shift in the coefficient. In all the examples that
we consider, the origin is a minimum if $\mathrm{Tr}(R) \leq 0$.
