group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
symmetric monoidal (∞,1)-category of spectra
Operads – or rather (∞,1)-operads such as given by topological operads or dendroidal sets – form an (∞,1)-category, and hence there is canonically a notion of cohomology associated with them, given as the sets of homotopy classes of morphisms out of a given operad into a given coefficient operad.
Explicitly this has been considered in (Rezk).