  We study the quantum fidelity approach to characterize thermal phase
transitions. Specifically, we focus on the mixed-state fidelity induced by a
perturbation in temperature. We consider the behavior of fidelity in two types
of second-order thermal phase transitions (based on the type of non-analiticity
of free energy), and we find that usual fidelity criteria for identifying
critical points is more applicable to the case of $\lambda$ transitions
(divergent second derivatives of free energy). Our study also reveals
limitations of the fidelity approach: sensitivity to high temperature thermal
fluctuations that wash out information about the transition, and inability of
fidelity to distinguish between crossovers and proper phase transitions. In
spite of these limitations, however, we find that fidelity remains a good
pre-criterion for testing thermal phase transitions, which we use to analyze
the non-zero temperature phase diagram of the Lipkin-Meshkov-Glick model.
