Here is a cohomology theory for a Hopf algebra, which I am sure has appeared elsewhere. I met it in the van Est spectral sequence for Hopf algebras. Apologies for my being stupid here, but it would be really helpful if someone would tell me where it comes from, and where to look for results on it!


Let $H$ be a Hopf algebra and $\lambda: F\to H\otimes
F$ be a left $H$-comodule, which we write in Sweedler notation $\lambda(f)=f\_{[-1]}\otimes f\_{[0]}$.


Define $D^{n}=H^{\otimes n+1}\otimes F$ for $n\ge 0$, with
the tensor product left $H$-coaction. The differential $ d:D^{n}\to D^{n+1}$ with $ d^2=0$ is defined by
 $$
 d(h\_{0}\otimes\ldots\otimes h\_{n}\otimes f)\,=\, \sum\_{n+1\ge i\ge
 0}(-1)^{i}\, h\_{0}\otimes\ldots\otimes h\_{i-1}\otimes 1\_H \otimes
 h\_{i}\otimes\ldots \otimes h\_{n}\otimes f\ .
 $$
As $ d$ is a left $H$-comodule map, we can restrict the complex to the invariants to give $({}^{coH}D^{n}, d)$, and the cohomology of the invariants is what I am looking for. The invariants are taken over the tensor product coaction on all factors. There is an explicit reformulation of the cohomology without taking invariants, and seemingly without requiring an algebra structure, though showing an isomorphism is awkward, and does require Hopf algebra structure:


Define a cochain complex
$(G^{\*},\bar d )$ by $G^{n}=H^{\otimes n}\otimes F$ for $n\ge 0$ with
 $\bar d f=1\_{H}\otimes f-\lambda(f)$ for $f\in F$ and
\begin{eqnarray\*}
 \bar d (h\_{1}\otimes\dots\otimes h\_{n}\otimes f) &=& 1\_{H}\otimes
h\_{1}\otimes\dots\otimes h\_{n}\otimes f\,-\, \Delta(h\_{1})\otimes\dots\otimes
h\_{n}\otimes f\,+\,\dots \cr &&+\,(-1)^{n}\,h\_{1}\otimes\dots\otimes
\Delta(h\_{n})\otimes f\,-\, (-1)^{n}\,h\_{1}\otimes\dots\otimes h\_{n}\otimes
\lambda(f)\ .
\end{eqnarray\*}
I guess that the comment below on the dependence on a coalgebra structure is then correct...



Identifying a Hopf algebra cohomology theory
Looks to me as if you have not used the product of your Hopf algebra, and it looks to me that you have written an example of a cotorsion product, as defined by Eilenberg and Moore in their paper Homology and fibrations I. Coalgebras, cotensor product and its derived functors. Comm. Math. Hel. 40(1965), 199--236, available here:
<http://retro.seals.ch/digbib/view?rid=comahe-002:1965-1966:40::223>


