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
A Wasserman. Equivariant differential topology, Topology 1969; 8(2):127-150.
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
MF Atiyah and R Bott. The Yang-Mills equations over Riemann surfaces. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences (1983): 523-615.
I personally found the exposition in Section 2 of Hingston's subsequent paper to be somewhat cleaner and more transparent:
N. Hingston. Equivariant Morse theory and closed geodesics. Journal of Differential Geometry 19 (1984), no. 1, 85--116.
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.
Austin, D. M.(1-IASP); Braam, P. J.(4-OX) Morse-Bott theory and equivariant cohomology. The Floer memorial volume, 123–183, Progr. Math., 133, Birkhäuser, Basel, 1995.