Equivariant generalized cohomology via stacks

We prove a general form of the statement that the cohomology of a quotient stack can be computed by the Borel construction. It also applies to the lisse extensions of generalized cohomology theories like motivic cohomology and algebraic cobordism. As a consequence, we deduce the localization property for the equivariant algebraic bordism theory of Deshpande-Krishna-Heller-Malag\'on-L\'opez. We also give a Bernstein-Lunts-type gluing description of the infinity-category of equivariant sheaves on a scheme X, in terms of nonequivariant sheaves on X and sheaves on its Borel construction.