General notions: Geometric cohomology, Arithmetic cohomology, Absolute cohomology. See these pages for more examples than those below. An important example is Mixed motives vs motivic cohomology.
One may ask for a better understanding of the appearance of Tannakian categories. Can one explain or express this in terms of homotopy theory? What happens over an arithmetic base? The notion of mixed motives (abelian) should make sense over an arithmetic base, so should have the Tannakian picture, but this setting is A1-invariant, so not good enough really.
Note that Weil-etale cohomology and Deninger cohomology might be an example. Are these homotopy invariant???
Continuous etale cohomology and Etale cohomology
nLab page on E40 Absolute and geometric theories