  We compute the stable cohomology groups of the mapping class groups of
compact orientable surfaces with one boundary, with twisted coefficients given
by the homology of the unit tangent bundle of the surface. This stable twisted
cohomology is not free as a module over the stable cohomology algebra with
constant coefficients. In fact, it is out of the scope of the traditional
framework for twisted cohomological stability, since these twisted coefficients
define a covariant functor over the classical category associated to mapping
class groups to study homological stability, rather than a contravariant one.
For comparison, we also compute the stable cohomology group with coefficients
in the first cohomology of the unit tangent bundle of the surface, which fits
into the traditional framework.
