  We describe a general procedure for computing renormalized curvature
integrals on Poincar\'e-Einstein manifolds. In particular, we explain the
connection between the Gauss-Bonnet-type formulas of Albin and Chang-Qing-Yang
for the renormalized volume, and explicitly identify a scalar conformal
invariant in the latter formula. Our approach constructs scalar conformal
invariants that are divergences at any Einstein manifold; these imply that the
scalar invariant in the Chang-Qing-Yang formula is not unique in dimension at
least eight. Our procedure also produces explicit conformally invariant
Gauss-Bonnet-type formulas for compact Einstein manifolds.
