  Four-dimensional asymptotically-free large $N$ gauge theories compactified on
$S^3_R \times \mathbb{R}$ have a weakly-coupled confining regime when $R$ is
small compared to the strong scale. We compute the vacuum energy of a variety
of confining large $N$ non-supersymmetric gauge theories in this calculable
regime, where the vacuum energy can be thought of as the $S^3$ Casimir energy.
The $N=\infty$ renormalized vacuum energy turns out to vanish in all of the
large $N$ gauge theories we have examined, confirming a striking prediction of
temperature-reflection symmetry.
