  Consider the tangent bundle of a Riemannian manifold $(M,g)$ of dimension
$n\geq3$ admitting a metric of negative curvature (not necessarily equal to
$g$) endowed with a twisted symplectic structure defined by a closed 2-form on
$M$. We consider the Hamiltonian flow generated (with respect to that
symplectic structure) by the standard kinetic energy Hamiltonian, and we
consider a compact regular energy level $\Sigma_{k}:=H^{-1}(k)$ of $H$. Suppose
$\Sigma_{k}$ is an Anosov energy level. We prove that if $n$ is odd, then if
the Hamiltonian flow restricted to $\Sigma_{k}$ is Anosov with $C^{1}$ weak
bundles then the Hamiltonian structure $(\Sigma_{k}$ is stable if and only if
it is contact. If $n$ is even and in addition the flow is assumed to be
1/2-pinched then the same conclusion holds. As a corollary we deduce that if
$g$ is negatively curved, strictly 1/4-pinched and the 2-form defining the
twisted symplectic structure is not exact then the Hamiltonian structure
$(\Sigma_{k}$ is never stable for all sufficiently large $k$.
