= Solution
Use the <canonical coframe of a surface unit tangent bundle> $(\alpha,\theta,\omega)$ and its dual $(X,H,V)$. In constant curvature $-1$, the decaying normal <Jacobi fields> have data $(j,j')=(a,-a)$, so the stable line is generated by $H-V$. The <weak stable bundle> is therefore
$$
E^{ws}=\operatorname{span}\{X,H-V\}.
$$
It is annihilated by the nonvanishing form $\zeta=\theta+\omega$. The structure equations now give
$$
d\theta=\alpha\wedge\omega,\qquad d\omega=\alpha\wedge\theta,
\qquad d\zeta=\alpha\wedge\zeta.
$$
Thus the auxiliary form in the <Godbillon-Vey invariant> can be taken as $\eta=\alpha$. In the natural orientation given by the oriented base and positive fibre rotation, put $\mu=\alpha\wedge\theta\wedge\omega$. Then
$$
\eta\wedge d\eta=\alpha\wedge\omega\wedge\theta=-\mu.
$$
This sign is important: in these conventions the contact three-form is the negative of the positive <Liouville volume of a surface geodesic flow>.
Integration around each circle fibre gives $\int_{S_xM}\omega=2\pi$, while $\alpha\wedge\theta=\pi^*\Omega_a$. Hence $\int_{SM}\mu=2\pi\operatorname{Area}(M)$. The <Gauss-Bonnet theorem> and $K=-1$ give $-\operatorname{Area}(M)=2\pi\chi(M)$. Consequently
$$
\boxed{\mathrm{gv}(E^{ws})=-2\pi\operatorname{Area}(M)=4\pi^2\chi(M)}.
$$
For genus $g\ge2$, this is $-8\pi^2(g-1)$ in the stated orientation. The opposite orientation of $SM$ reverses the sign.
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