In this paper, we consider piecewise smooth dynamical systems with generalized indifferent periodic orbits associated to a given potential function. We show that the presence of such orbits causes non-uniqueness of equilibrium states (phase transitions) and non-Gibbsianness of equilibrium measures. If, in addition, the system exhibits local exponential instability, we show that the distribution of the stopping times over hyperbolic regions has powerlike tails with respect to any s-conformal measure, where s is the smallest zero of Bowen's equation. Finally, we prove that the Hausdorff dimension of the limit set is equal to s provided the dynamics is piecewise conformal.