If A is a commutative ring spectrum one can form X \otimes A for X a (finite pointed) simplicial set. Important examples are topological Hochschild homology (X = S^1), torus homology (X = (S^1)^n) or higher THH (X = S^n). In joint work with Ayelet Lindenstrauss we investigate the question when X \otimes A only depends on the homotopy type of the suspension, \Sigma X. I will present counterexamples but also classes of examples that are stable in the above sense.