Let $k$ be a field of characteristic $0$ and $A=k\langle\langle x_1,\dotsc,x_n\rangle\rangle$ be a free completed associative algebra. The space of continuous derivations $\mathrm{Der}(A)$ is isomorphic, as a vector space, to $A^{\oplus n}$. $\mathrm{Der}(A)$ is a non-commutative analogue of the space of formal vector fields on an $n$-dimensional space.
Question: How can we compute its Lie algebra cohomology $H^\bullet(\mathrm{Der}(A),M)$ for some coefficient $M$?
We may take $M$ as $A$ itself, the cyclic quotient $A/[A,A]$, the enveloping algebra $A^e$ or something else.
In commutative case (including $n=1$ in above), they are pretty much solved in Cohomology of Infinite-dimensional Lie algebras by D. B. Fuks (for example, see Theorem 2.2.7). I also looked up Kontsevich's Formal (non-)commutative symplectic geometry, but nothing there particularly helped.
Any comments or references are appreciated. This is a duplicate of my question on MO.