  We give a recipe for computing correlation functions of the displacement
operator localized on a spherical or planar higher dimensional twist defect
using AdS/CFT. Such twist operators are typically used to construct the $n$'th
Renyi entropies of spatial entanglement in CFTs and are holographically dual to
black holes with hyperbolic horizons. The displacement operator then tells us
how the Renyi entropies change under small shape deformations of the entangling
surface. We explicitly construct the bulk to boundary propagator for the
displacement operator insertion as a linearized metric fluctuation of the
hyperbolic black hole and use this to extract the coefficient of the
displacement operator two point function $C_D$ in any dimension. The $n
\rightarrow 1$ limit of the twist displacement operator gives the same bulk
response as the insertion of a null energy operator in vacuum, which is
consistent with recent results on the shape dependence of entanglement entropy
and modular energy.
