  We study the holographic entanglement entropy of deformed entangling regions
in three-dimensional CFTs dual to Einstein-AdS gravity, using a renormalization
scheme based on the addition of extrinsic counterterms. In this prescription,
when even-dimensional manifolds are considered, the universal contribution to
the entanglement entropy is identified as the renormalized volume of the
Ryu-Takayanagi hypersurface, which is written as the sum of a topological and a
curvature term. It is shown that the change in the renormalized entanglement
entropy due to the deformation of the entangling surface is encoded purely in
the curvature contribution. In turn, as the topological part is given by the
Euler characteristic of the Ryu-Takayanagi surface, it remains shape
independent. Exploiting the covariant character of the extrinsic counterterms,
we apply the renormalization scheme for the case of deformed entangling regions
in AdS$_4$/CFT$_3$, recovering the results found in the literature. Finally, we
provide a derivation of the relation between renormalized entanglement entropy
and Willmore energy. The presence of a lower bound of the latter makes manifest
the relation between the AdS curvature of the Ryu-Takayanagi surface and the
strong subadditivity property.
