Let $G$ be an algebraic group. Consider the classifying stack $\left[\text{pt}/G\right]$, where $\text{pt} = \text{Spec } \mathbb{C}$.
What's the cohomology of the classifying stack $\left[\text{pt}/G\right]$ ?
Here, by the cohomology of $\left[\text{pt}/G\right]$, I mean the singular cohomology ring of $\left[\text{pt}/G\right]$ with coefficients in $\mathbb{Q}$, denoted by $H^*\left(\left[\text{pt}/G\right],\mathbb{Q}\right)$. We know the coarse moduli space of the classifying stack $\left[\text{pt}/G\right]$ is just the one-point space $\text{pt}$. I'm not sure if $H^*\left(\left[\text{pt}/G\right],\mathbb{Q}\right)$ is defined as $H^*\left(\text{pt},\mathbb{Q}\right)$ or $H^*_{G}\left(\text{pt},\mathbb{Q}\right)$, i.e., the $G$-equivariant cohomology of a point.
Edit 1
The algebraic group $G$ may be discrete or continuous. I'm mainly interested in two cases: 1) $G = (\mathbb{C}^*)^n$, and 2) $G = \mathbb{Z}/n\mathbb{Z}$, where $n$ is some positive integer.