A comment on curvature effects in CFTs and the Cardy–Verlinde formula

We examine the Cardy–Verlinde formula for finite temperature N=4 super-Yang–Mills theory on R×S3, and its AdS dual. We find that curvature effects introduce non-trivial corrections to thermodynamic quantities computed on both sides. We find a modified version of the Cardy–Verlinde formula for the SYM theory, incorporating these. On the gravity side, these corrections imply that the Cardy–Verlinde formula is exact.