  We study the index of $\mathcal{N}=4$ Yang-Mills theory on
$S^3\times\mathbb{R}$. We argue that the index should undergo a large $N$
deconfinement phase transition, by computing an upper bound of its
`temperature.' We compute this bound by optimizing the phases of fugacities.
The bound we find has some features analogous to the Hagedorn temperature. We
briefly discuss a possible mechanism of the actual deconfinement transition
below our bound. Our upper bound is lower than the Hawking-Page transition
`temperature' of known BPS black holes in the AdS$_5$ dual. We thus expect the
existence of new black holes.
