Let $M$ be a compact smooth manifold. And particularly I am interested in the case the torus $M=T^n$.


Consider the de Rham complex $(\Omega^\*(M), d)$ and the cochain complex
$$
C:=\mathrm{Hom} (\Omega^\*(M),\Omega^\*(M))
$$
with the differential map $\delta$ given by
$$
\delta(f)=d \circ f - f\circ d$$



> 
> **Question**: what is the cohomology $H^\*(C,\delta)$? And, what should be the right tool to do the computation? Is this cohomology related to the usual de Rham cohomology $H^\*\_{dR}(M)$?
> 
> 
> 



Compute the cohomology of $\mathrm{Hom} (\Omega^*(M),\Omega^*(M))$
Since everything is linear over the field $\mathbb{R}$, $\Omega^\*(M)$ is chain homotopy equivalent to $H^\*\_{dR}(M)$ endowed with the trivial differential, hence the cohomology of your complex $C$ is just $\operatorname{Hom}(H^\*\_{dR}(M),H^\*\_{dR}(M))$. This identification is not natural in $M$ though.


