It is well-known the next theorem at Chapter III, Theorem 7.2. in Bredon's Introduction to compact transformation groups book.
Theorem: Let $X$ be a paracompact $G$-space with $G$ finite and let $\pi:X \rightarrow X/G$ be the orbit map. If $k$ is a field of characteristic zero, then $$\pi^*:H^*(X/G;k) \rightarrow H^*(X;k)^G$$ is an isomorphism, where $H^*(X;k)^G$ is the fixed point set of $G$ action on $H^*(X;k)$.
My Question: If $G$ is a compact, totally disconnected group and $X$ is a paracompact $G$-space, then $$\pi^*:H^*(X/G;k) \rightarrow H^*(X;k)^G$$ is an isomorphism?