  We compute the Poisson cohomology of the one-parameter family of
SU(2)-covariant Poisson structures on the homogeneous space
S^{2}=CP^{1}=SU(2)/U(1), where SU(2) is endowed with its standard Poisson--Lie
group structure,thus extending the result of Ginzburg \cite{Gin1} on the
Bruhat--Poisson structure which is a member of this family. In particular, we
compute several invariants of these structures, such as the modular class and
the Liouville class. As a corollary of our computation, we deduce that these
structures are nontrivial deformations of each other in the direction of the
standard rotation-invariant symplectic structure on S^{2}; another corollary is
that these structures do not admit smooth rescaling.
