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Given a conformally variational scalar Riemannian invariant $I$, we identifya sufficient condition for a closed Riemannian manifold to admit finite regularcoverings with many nonhomothetic conformal rescalings with $I$ constant. Wealso identify a sufficient condition for the universal cover to admitinfinitely many geometrically distinct periodic conformal rescalings with $I$constant. Using these conditions, we improve known nonuniqueness results forthe $Q$-curvatures of orders two, four, and six, and establish nonuniquenessresults for higher-order $Q$-curvatures and renormalized volume coefficients.

A general nonuniqueness result for Yamabe-type problems for conformally variational Riemannian invariants