The article Deformation par quantification et rigidite des algebres enveloppantes 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.

The article Deformation par quantification et rigidite des algebres enveloppantes 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 by Whitehead's second Lemma for Lie algebra cohomology.

The article Deformation par quantification et rigidite des algebres enveloppantes 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.

The article Deformation par quantification et rigidite des algebres enveloppantes 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.