  For a class of non-uniformly hyperbolic interval maps, we study rates of
escape with respect to conformal measures associated with a family of geometric
potentials. We establish the existence of physically relevant conditionally
invariant measures and equilibrium states and prove a relation between the rate
of escape and pressure with respect to these potentials. As a consequence, we
obtain a Bowen formula: we express the Hausdorff dimension of the set of points
which never exit through the hole in terms of the relevant pressure function.
Finally, we obtain an expression for the derivative of the escape rate in the
zero-hole limit.
