Put in other words, given an evendimensional sphere $S^{2k}$: is there a manifold $M$ such that $T^* M$ is diffeomorphic to $S^{2k}$?

Of course, the spheres are compact while cotangent bundles are noncompact (unless in dimension 0). Nevertheless, a bit more interesting is the question whether the even dimensional spheres can be phase spaces in the sense of symplectic manifolds. There the $\mathbb{S}^2$ is an example: the volume form is nondegenerate and a twoform. Closedness is for free in 2 dimensions. The higher dimensional spheres $\mathbb{S}^{2n}$ are never symplectic as on a compact symplectic manifold, the deRham cohomology has to be sufficiently nontrivial: the class of the symplectic form and all its $\wedge$powers up to $n$ are nontrivial. For $\mathbb{S}^{2n}$ and $n \ge 2$ this is known to be not true: all cohomologies vanish except for the zeroth and the $2n$th, which are both onedimensional. 


To answer the question in the title: if by phase space we mean a symplectic manifold, then only for $k=1$ is there a symplectic structure. This is the phase space of a classical spin. It is not necessary for a manifold to be identified with $T^*M$ for some $M$ to qualify as a phase space. This is the first place we encounter the idea, with $M$ being the configuration space of a system, but the concept is more general 

