A generalization of the Connes-Thom isomorphism is given for stable, homotopy invariant, and split exact functors on separable
C
∗
C^*
-algebras. As examples of these functors, we concentrate on asymptotic and local cyclic cohomology, and the result is applied to improve some formulas in asymptotic and local cyclic cohomology of
C
∗
C^*
-algebras. As another application, it is shown that these cyclic theories are rigid under Rieffel’s deformation quantizations.