There's been a good deal of work since the papers Mark Grant cited.
The rational cohomology of $\mathrm{Hom}(\mathbb{Z}^n,K)//K$ for $K$ a compact connected Lie group was computed by Stafa (https://arxiv.org/abs/1705.01443). It's a theorem of Florentino and Lawton that if $G$ is a linearly reductive Lie group with maximal compact subgroup $K$, then $\mathrm{Hom}(\mathbb{Z}^n,G)//G$ deformation retracts to $\mathrm{Hom}(\mathbb{Z}^n,K)/K$, so for the general linear group, we can switch to working with the unitary groups instead. Stafa gives a general formula for the Poincare series, in terms of the order of the Weyl group and its action on the (dual of the) Lie algbra of a maximal torus. The formula reduces to $((1+t)^{2n}+(1-t^2)^n)/2$ for $G = U(n)$ (or $GL_n (\mathbb{C})$). Florentino and Silva (https://arxiv.org/abs/1711.07909) computed algebro-geometric refinements of these Poincare series, and their work recovers Stafa's formula.
There's a similar story for the ordinary rational cohomology of $\mathrm{Hom}(\mathbb{Z}^n,G)$, discussed in a paper I wrote with Stafa, https://arxiv.org/abs/1704.05793. Since then D. Kishimoto and M. Takeda have made a good deal of progress, including information about the ring structure and torsion (also they gave a much shorter derivation of the Poincare series).
Regarding equivariant cohomology, Baird has some work in the compact case; see Section 4 of his paper https://arxiv.org/abs/math/0610761. Note that the inclusion of $\mathrm{Hom}(\mathbb{Z}^n,K)$ into $\mathrm{Hom}(\mathbb{Z}^n,G)$ is $K$-equivariant, and the inclusion of $K$ into $G$ is a homotopy equivalence, so $H_K^* (\mathrm{Hom}(\mathbb{Z}^n,K) \cong H_K^* (\mathrm{Hom}(\mathbb{Z}^n,G)) \cong H_G^* \mathrm{Hom}(\mathbb{Z}^n,G)$. It would be quite interesting to know more about the equivariant cohomology.