  We show that for a d-dimensional CFT in flat space, the Renyi entropy S_q
across a spherical entangling surface has the following property: in an
expansion around q=1, the first correction to the entanglement entropy is
proportional to C_T, the coefficient of the stress tensor vacuum two-point
function, with a fixed d-dependent coefficient. This is equivalent to a similar
statement about the free energy of CFTs living on S^1 x H^{d-1} with inverse
temperature \beta=2\pi q. In addition to furnishing a direct argument
applicable to all CFTs, we exhibit this result using a handful of gravity and
field theory computations. Knowledge of C_T thus doubles as knowledge of Renyi
entropies in the neighborhood of q=1, which we use to establish new results in
3d vector models at large N.
