  We study a system of non-interacting active particles, propelled by colored
noises, characterized by an activity time $\tau$, and confined by a double-well
potential. A straightforward application of this system is the problem of
barrier crossing of active particles, which has been studied only in the limit
of small activity. When $\tau$ is sufficiently large, equilibrium-like
approximations break down in the barrier crossing region. In the model under
investigation, it emerges a sort of "negative temperature" region, and
numerical simulations confirm the presence of non-convex local velocity
distributions. We propose, in the limit of large $\tau$, approximate equations
for the typical trajectories which successfully predict many aspects of the
numerical results. The local breakdown of detailed balance and its relation
with a recent definition of non-equilibrium heat exchange is also discussed.
