  Renormalized entanglement entropy can be defined using the replica trick for
any choice of renormalization scheme; renormalized entanglement entropy in
holographic settings is expressed in terms of renormalized areas of extremal
surfaces. In this paper we show how holographic renormalized entanglement
entropy can be expressed in terms of the Euler invariant of the surface and
renormalized curvature invariants. For a spherical entangling region in an
odd-dimensional CFT, the renormalized entanglement entropy is proportional to
the Euler invariant of the holographic entangling surface, with the coefficient
of proportionality capturing the (renormalized) F quantity. Variations of the
entanglement entropy can be expressed elegantly in terms of renormalized
curvature invariants, facilitating general proofs of the first law of
entanglement.
