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I have added (here) some more lead-in words for the equivariant case (but still thin) and then I started to spell out (here) detail/proof for the claims implicit in FHT 07, (3.5).
(What I don’t see yet is how to prove that the action of on the cohomology of the fiber is trivial, so that their Serre spectral sequence really applies – have left a comment on this here.)
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A spectral sequence for twisted de Rham cohomology is discussed in
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Pierre Deligne, Section II.6 in: Equations différentielles à points singuliers réguliers, Lecture Notes in Math 163, Springer 1970 (pdf, publication.ias:355)
Pavel I. Etingof, Igor Frenkel, Alexander A Kirillov, Section 7.2 in: Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, Mathematical surveys and monographs 58, American Mathematical Society (1998)
for textbook/moograph discussion of 1-twisted de Rham cohomology.
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