Renormalizing curvature integrals on Poincaré–Einstein manifolds

After analyzing renormalization schemes on a Poincaré–Einstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is well known and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms and their behavior under a variation of the Poincaré–Einstein structure and obtain, from the renormalized integral of the Pfaffian, an extension of the Gauss–Bonnet theorem.