  We elaborate the description of the semi-classical gravity sector of
Yang-Mills matrix models on a covariant quantum FLRW background. The basic
geometric structure is a frame, which arises from the Poisson structure on an
underlying $S^2$ bundle over space-time. The equations of motion for the
associated Weitzenb\"ock torsion obtained in arXiv:2002.02742 are rewritten in
the form of Yang-Mills-type equations for the frame. An effective action is
found which reproduces these equations of motion, which contains an
Einstein-Hilbert term coupled to a dilaton, an axion and a Maxwell-type term
for the dynamical frame. An explicit rotationally invariant solution is found,
which describes a gravitational field coupled to the dilaton.
