Lubin and Tate, in discussing moduli of 1-dimensional formal groups construct a cohomology theory of formal groups, at least in degrees 0,1 and 2. Does their result about deformations actually follow from the natural 3rd group of this cohomology being trivial? Has the obvious extension of this structure to a full cohomology theory been written down somewhere and had its properties studied? Moreover, Lubin and Tate's cohomology actually appears to be a special a case of what one might call the Hochschild cohomology of $\hat{\mathbb{G}}_a$ (using Demazure and Gabriel's notion of Hochschild cohomology of group schemes) with coordinates in some other formal group $\mathbb{G}$ (the one defining the group law of interest) with a trivial action of $\hat{\mathbb{G}}_a$ on $\mathbb{G}$. The same structure arises in Lazard's proof of the extensibility of formal group n-buds. Has this interpretation been written down anywhere? And more importantly, are there other interesting examples of applications of this sort of structure?
Looking at this question causes me to think I might find answers in the manuscript of Strickland's which is online, but I'm not sure yet.
