  We study the stable cohomology groups of the mapping class groups of surfaces
with twisted coefficients given by the $d^{th}$ exterior powers of the first
rational homology of the unit tangent bundles of the surfaces
$\tilde{H}_{\mathbb{Q}}$. These coefficients are outside of the traditional
framework of cohomological stability. They form a module
$H_{\mathrm{st}}^{*}(\Lambda^{d}\tilde{H}_{\mathbb{Q}})$ over the stable
cohomology algebra of the mapping class groups with trivial coefficients
denoted by $\mathrm{Sym}_{\mathbb{Q}}(\mathcal{E})$. If $d\neq 2$, the
$\mathrm{Tor}$-group in each degree of
$H_{\mathrm{st}}^{*}(\Lambda^{d}\tilde{H}_{\mathbb{Q}})$ does not vanish, and
we compute all these $\mathrm{Tor}$-groups explicitly for $d \leq 5$. In
particular, for each $d\neq 2$, the module
$H_{\mathrm{st}}^{*}(\Lambda^{d}\tilde{H}_{\mathbb{Q}})$ is not free over
$\mathrm{Sym}_{\mathbb{Q}}(\mathcal{E})$, while it is free for $d=2$.For
comparison, we also compute the stable cohomology group with coefficients in
the $d^{th}$ exterior powers of the first rational cohomology of the unit
tangent bundle of the surface, which fit into the classical framework of
cohomological stability.
