  We derive necessary and sufficient conditions for large $N$ conformal field
theories to have a universal free energy and an extended range of validity of
the higher-dimensional Cardy formula. These constraints are much tighter than
in two dimensions and must be satisfied by any conformal field theory dual to
Einstein gravity. We construct and analyze symmetric product orbifold theories
on $\mathbb{T}^d$ and show that they only realize the necessary phase structure
and extended range of validity if the seed theory is assumed to have a
universal vacuum energy.
