  We present a holographic method for computing the response of R\'enyi
entropies in conformal field theories to small shape deformations around a flat
(or spherical) entangling surface. Our strategy employs the stress tensor
one-point function in a deformed hyperboloid background and relates it to the
coefficient in the two-point function of the displacement operator. We obtain
explicit numerical results for $d=3,\dots,6$ spacetime dimensions, and also
evaluate analytically the limits where the R\'enyi index approaches 1 and 0 in
general dimensions. We use our results to extend the work of 1602.08493 and
disprove a set of conjectures in the literature regarding the relations between
the R\'enyi shape dependence and the conformal weight of the twist operator. We
also extend our analysis beyond leading order in derivatives in the bulk theory
by studying Gauss-Bonnet gravity.
