Let $\hat{\mathbb G}$ be a formal group over a field $k$, and let $V$ be a finite dimensional algebraic representation of $\hat{\mathbb G}$ (meaning we have fixed a homomorphism of algebraic groups $\rho:\hat{\mathbb G}\to\operatorname{GL}(V)$ over $k$).
I would like to consider the rational cohomology of $\hat{\mathbb G}$ with coefficients in $V$, denoted $H^\ast(\hat{\mathbb G},V)$. By definition, this is the cohomology of the complex whose $p$-cochains are the additive group of regular functions $$f:\overbrace{\hat{\mathbb G}\times\cdots\times\hat{\mathbb G}}^{p\text{ times}}\to V$$ and whose differential is given by the usual formula $$(\delta f)(t_1,\ldots,t_p)=\rho(t_1)f(t_2,\ldots,t_p)-f(t_1t_2,t_3,\ldots,t_p)+f(t_1,t_2t_3,t_4,\ldots,t_p)-\cdots$$ I presume much must be known about the resulting cohomology groups $H^\ast(\hat{\mathbb G},V)$, but I do not know where to look.
I believe I can show that $H^\ast(\hat{\mathbb G}_{\mathrm{m}},k_{\mathrm{std}}^{\otimes n})=0$ for integers $n\ne 0$, where $\hat{\mathbb G}_{\mathrm{m}}$ is the formal multiplicative group and $k_{\mathrm{std}}$ is its standard representation. Is this a known result for which there exists a canonical reference? It looks like I need this result for some specific application, and I'd like to get a feeling for what is known in this area.