group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The term $p$-adic cohomology would broadly refer to cohomology over spaces in p-adic geometry.
More specifically it is used for crystalline cohomology in positive characteristic, based on the comparison theorem which says that this is equivalent to de Rham cohomology in p-adic geometry.
Notice that by “$p$-adic cohomology” one could also mean cohomology with coefficients in p-adic numbers. An example of this is ℓ-adic cohomology which has ℓ-adic coefficients, genuinely so over the pro-étale site). But not in all cases does this give the expected answers.