Consider topological group $\mathbb{R}^\times$ with a standard topology and $\mathbb{R}^\times$-module $S'(\mathbb{R})$ consisting of all Schwartz distributions on $\mathbb{R}$ where $\alpha\in\mathbb{R}^\times$ acts by change of variable: $u(x) \mapsto u(\alpha x)$. The module $S'(\mathbb{R})$ is an infinite-dimensional topological vector space over $\mathbb{C}$.
The $0$'th cohomology group $H^0(\mathbb{R}^\times,S'(\mathbb{R}))$ is one-dimensional, and consists of constant functions, which are distributions invariant under the change of variable.
What about higher $H^k(\mathbb{R}^\times,S'(\mathbb{R}))$? Are they trivial? Finite dimensional? How one can calculate them?