Stability of Loday constructions

We study the question for which commutative ring spectra $A$ the tensor of a simplicial set $X$ with $A$, $X\otimes A$, is a stable invariant in the sense that it depends only on the homotopy type of $\mathrm{\Sigma }X$. We prove several structural properties about different notions of stability, corresponding to different levels of invariance required of $X\otimes A$, and establish stability in important cases, such as complex and real periodic topological K-theory, $KU$ and $KO$.