Amalendu Krishna has given talks on stacks and their cohomology theories (in Oslo).
See the stacks project, and stuff of Joshua maybe.
If we use simplicial sheaves, can we get automatic extension of (some) CTs to (some notion of) stacks?
Bredon style cohomology of stacks, Chow groups of stacks, G-theory of algebraic stacks, K-theory of stacks, Motivic cohomology of stacks, Stack cohomology
nLab page on E70 Cohomology theories for stacks