  In this paper we introduce the notion of rigidity for harmonic-Ricci solitons
and we provide some characterizations of rigidity, generalizing some known
results for Ricci solitons. In the compact case we are able to deal with not
necessarily gradient solitons while, in the complete non-compact case, we
restrict our attention to steady and shrinking gradient solitons. We show that
the rigidity can be traced back to the vanishing of certain modified curvature
tensors that take into account the geometry a Riemannian manifold equipped with
a smooth map $\varphi$, called $\varphi$-curvatures, which are a natural
generalization of the standard curvature tensors in the setting of
harmonic-Ricci solitons.
