  One defines the notion of universal deformation quantization: given any
manifold $M$, any Poisson structure $\P$ on $M$ and any torsionfree linear
connection $\nabla$ on $M$, a universal deformation quantization associates to
this data a star product on $(M,\P)$ given by a series of bidifferential
operators whose corresponding tensors are given by universal polynomial
expressions in the Poisson tensor $\P$, the curvature tensor $R$ and their
covariant iterated derivatives. Such universal deformation quantization exist.
We study their unicity at order 3 in the deformation parameter, computing the
appropriate universal Poisson cohomology.
