Let $X_0$ be some algebro-geometric object defined over a field, and suppose its deformation functor is prorepresentable, so there is a universal deformation ring $R$. Then $Aut(X_0)$ acts naturally on the deformation functor, and hence on $R$. One may consider the group cohomology $$H^i(Aut(X_0),R)$$ where $R$ is considered as its underlying additive group.

Do these cohomology groups have a "geometric interpretation"?