Is there some variant on Morse or Morse-Bott theory yielding equivariant (co)homology instead of singular homology?
Any reference/idea would be greatly appreciated.
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The answer to your question is yes, of course. The theory has been around at least since the late 60s! See Wasserman's paper
I think the first "big application" is due to Atiyah and Bott, who used it to study (what else?) Yang-Mills on surfaces. See the first section of this 90 page behemoth
I personally found the exposition in Section 2 of Hingston's subsequent paper to be somewhat cleaner and more transparent:
I'm sure this equivariant Morse theory is in many textbooks by now, but the original papers cover much of what it is and how it is used. There's even a recent discrete version for $G$-simplicial complexes due to Freij!
If you like a Morse homological treatment, you could also look at the paper of Austin and Braam.