The article [Deformation par quantification et rigidite des
algebres enveloppantes](https://arxiv.org/abs/math/0211416v1) by M. Bordemann, A. Makhlouf, T. Petit addresses these questions. They call Lie algebras $\mathfrak{g}$ with $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))=0$ *strongly rigid*, and show that then
every formal associative deformation is equivalent to the trivial deformation. For semisimple Lie algebras over an algebraically closed field of characteristic zero one has $HH^2(U(\mathfrak{g}),U(\mathfrak{g}))=H^2(\mathfrak{g},S\mathfrak{g})$, which is zero.