Higher Hochschild homology is not a stable invariant

Higher Hochschild homology is the analog of the homology of spaces, where the context for the coefficients – which usually is that of abelian groups – is that of commutative algebras. Two spaces that are equivalent after a suspension have the same homology. We show that this is not the case for higher Hochschild homology, providing a counterexample to a behavior so far observed in stable homotopy theory.