  We develop a framework for studying the well-known universal term in the
Renyi entropy for an arbitrary entangling region in four-dimensional conformal
field theories that are holographically dual to gravitational theories. The
shape dependence of the Renyi entropy $S_n$ is described by two coefficients:
$f_b(n)$ for traceless extrinsic curvature deformations and $f_c(n)$ for Weyl
tensor deformations. We provide the first calculation of the coefficient
$f_b(n)$ in interacting theories by relating it to the stress tensor one-point
function in a deformed hyperboloid background. The latter is then determined by
a straightforward holographic calculation. Our results show that a previous
conjecture $f_b(n) = f_c(n)$, motivated by surprising evidence from a variety
of free field theories and studies of conical defects, fails holographically.
