This is an expansion to Oscar's answer. The cited result on the cohomology with rational coefficients is an application of the Transfer Theorem:

> Let $G$ be a finite group, $X$ a topological manifold and $F$ a field with 
> $\mbox{char}\,F = 0$ or $\mbox{char}\,F \nmid o(G)$, then
> $$H^\*(X/G; F) \cong H^*(X;F)^G$$

There is a nice proof of this theorem on Bredon's book "Introduction to compact transformation groups."

For the case of $\mathbb R^k$ you can compute the cohomology (with complex coefficients) explicitly. The answer depends on the parity of $k$. For odd $k$, $H^*(F_n(\mathbb{R}^k); \mathbb{C})$ is one dimensional in degree $0$ and trivial otherwise, for even $k$ it is one-dimensional in degrees $0$ and $k-1$ and trivial otherwise.