  A novel gravity theory based on Poisson Generalized Geometry is investigated.
A gravity theory on a Poisson manifold equipped with a Riemannian metric is
constructed from a contravariant version of the Levi-Civita connection, which
is based on the Lie algebroid of a Poisson manifold. Then, we show that in
Poisson Generalized Geometry the $R$-fluxes are consistently coupled with such
a gravity. An $R$-flux appears as a torsion of the corresponding connection in
a similar way as an $H$-flux which appears as a torsion of the connection for-
mulated in the standard Generalized Geometry. We give an analogue of the
Einstein-Hilbert action coupled with an $R$-flux, and show that it is invariant
under both $\beta$-diffeomorphisms and $\beta$-gauge transformations.
