  After analyzing renormalization schemes on a Poincar\'e-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 Poincar\'e-Einstein structure, and obtain,
from the renormalized integral of the Pfaffian, an extension of the
Gauss-Bonnet theorem.
