  In this paper we study the phenomenon of phase transitions for the geodesic
flow on some geometrically finite negatively curved manifolds. We define a
class of potentials going slowly to zero through the cusps of $M$ for which the
pressure map exhibits a phase transition. By a careful choice of the metric at
the cusp we construct a geometrically finite manifold for which the geometric
potential (or unstable Jacobian) exhibits a phase transition. Our results
apply, in particular, to the geodesic flow on an $M$-puncture sphere, for every
$M\ge 3$, and a suitable choice of Riemannian metric.
