  Two-dimensional conformal field theories exhibit a universal free energy in
the high temperature limit $T \to \infty$, and a universal spectrum in the
Cardy regime, $\Delta \to \infty$. We show that a much stronger form of
universality holds in theories with a large central charge $c$ and a sparse
light spectrum. In these theories, the free energy is universal at all values
of the temperature, and the microscopic spectrum matches the Cardy entropy for
all $\Delta \geq c/6$. The same is true of three-dimensional quantum gravity;
therefore our results provide simple necessary and sufficient criteria for 2d
CFTs to behave holographically in terms of the leading spectrum and
thermodynamics. We also discuss several applications to CFT and gravity,
including operator dimension bounds derived from the modular bootstrap,
universality in symmetric orbifolds, and the role of non-universal `enigma'
saddlepoints in the thermodynamics of 3d gravity.
