  In this paper, we derive a general pressure gap criterion for closed rank 1
manifolds whose singular sets are given by codimension 1 totally geodesic flat
subtori. As an application, we show that under certain curvature constraints,
potentials that decay faster than geometric potentials (towards to the singular
set) have pressure gaps and have no phase transitions. Along the way, we prove
that geometric potentials are H\"older continuous near singular sets.
