  We extend to finite temperature the fidelity approach to quantum phase
transitions (QPTs). This is done by resorting to the notion of mixed-state
fidelity that allows one to compare two density matrices corresponding to two
different thermal states. By exploiting the same concept we also propose a
finite-temperature generalization of the Loschmidt echo. Explicit analytical
expressions of these quantities are given for a class of quasi-free fermionic
Hamiltonians. A numerical analysis is performed as well showing that the
associated QPTs show their signatures in a finite range of temperatures.
