  We study geodesic flows over compact rank 1 manifolds and prove that
sufficiently regular potential functions have unique equilibrium states if the
singular set does not carry full pressure. In dimension 2, this proves
uniqueness for scalar multiples of the geometric potential on the interval
$(-\infty,1)$, which is optimal. In higher dimensions, we obtain the same
result on a neighborhood of 0, and give examples where uniqueness holds on all
of $\mathbb{R}$. For general potential functions $\varphi$, we prove that the
pressure gap holds whenever $\varphi$ is locally constant on a neighborhood of
the singular set, which allows us to give examples for which uniqueness holds
on a $C^0$-open and dense set of H\"older potentials.
