Definition
Suppose that an algebraic (Lie) group acts on a scheme (manifold) \(X\). Then \[\mathbb EG\times^GX:= (\mathbb EG\times X)/(e\cdot g, x)\sim (e, gx)\] and we define the \(i\)th equivariant cohomology of \(X\) to be \[H^i_GX:=H^i(\mathbb EG\times ^GX).\]
When \(X\) is a scheme and \(\mathbb EG\) doesn’t make sense, you can either use a stacky construction or approximation spaces.