  Let $\varphi$ and $\varphi'$ be two homotopic actions of the topological
group $G$ on the topological space $X$. To an object $A$ in the $G$-equivariant
derived category $D_{\varphi}(X)$ of $X$ relative to the action $\varphi$ we
associate an object $A'$ of category $D_{\varphi'}(X)$, such that the
corresponding $G$-equivariant compactly supported cohomologies $H_{G,c}(X,\,A)$
and $H_{G,c}(X,\,A')$ are isomorphic. When $G$ is a Lie group and $X$ is a
subanalytic space, we prove that the $G$-equivariant cohomologies
$H_{G}(X,\,A)$ and $H_{G}(X,\,A')$ are also isomorphic.
