group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $X$ a smooth manifold and $H \in \Omega^3(X)$ a closed differential 3-form, the $H$ twisted de Rham complex is the $\mathbb{Z}_2$-graded vector space $\Omega^{even}(X) \oplus \Omega^{odd}(X)$ equipped with the $H$-twisted de Rham differential
Notice that this is nilpotent, due to the odd degree of $H$, such that $H \wedge H = 0$, and the closure of $H$, $d H = 0$.
There is also the cohomology of the chain complex whose maps are just multiplication by $H$.
This is also called H-cohomology (Cavalcanti 03, p. 19).
If the de Rham complex $(\Omega^\bullet(X),d_{dr})$ is formal, then for $H \in \Omega_{cl}^3(X)$ a closed differential 3-form, the $H$-twisted de Rham cohomology of $X$ coincides with its H-cohomology, for any closed 3-form $H$.