As in the question For a G-variety, what could one say about the motif of the corresponding simplicial variety I am in the following situation: $G$ is an algerbraic group, and X is a smooth $G$-variety. Then the simlicial variety whose terms are $G^i\times X$ yields a 'complex of varieties' $\dots\to G\times G\times X\to G\times X\to X$; next I obtain a Voevodsky's motivic complex $C$.
Next, I want to compute the (Zariski) cohomology of $C$ with coefficients in $G_m$. How could one do this? David Ben-Zvi wrote that this is the motif of the stack $X/G$. I don't know anything about stacks, so I would like to know whether this helps. For example, can one prove that $H^2(C,G_m)$ vanishes?