We study the classifying space of a twisted loop group L_σG where G is a compact Lie group and σ is an automorphism of G of finite order modulo inner automorphisms. Equivalently, we study the σ-twisted adjoint action of G on itself. We derive a formula for the cohomology ring H^*(BL_σG) and explicitly carry out the calculation for all automorphisms of simple Lie groups. More generally, we derive a formula for the equivariant cohomology of compact Lie group actions with constant rank stabilizers.