group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
This entry collects links related to the book
on parameterized stable homotopy theory, hence on stable homotopy theory in slice (infinity,1)-toposes Top$/X$ for given topological base spaces $X$: the homotopy theory of ex-spaces and parametrized spectra.
A survey is in the slides
One application is twisted cohomology: instead of cocycles given by maps $X \to A$, twisted cocycles are given by sections $X \to P$ of a bundle $P \to X$ of spectra over $X$.
A discussion of some of these issues using tools from (infinity,1)-category theory are in
See also
A general abstract context for parameterized spectra are tangent (infinity,1)-toposes.