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...