  We study finite-temperature Dynamical Quantum Phase Transitions (DQPTs) by
means of the fidelity and the interferometric Loschmidt Echo (LE) induced
metrics. We analyse the associated dynamical susceptibilities (Riemannian
metrics), and derive analytic expressions for the case of two-band
Hamiltonians. At zero temperature the two quantities are identical,
nevertheless, at finite temperatures they behave very differently. Using the
fidelity LE, the zero temperature DQPTs are gradually washed away with
temperature, while the interferometric counterpart exhibits finite-temperature
Phase Transitions (PTs). We analyse the physical differences between the two
finite-temperature LE generalisations, and argue that, while the
interferometric one is more sensitive and can therefore provide more
information when applied to genuine quantum (microscopic) systems, when
analysing many-body macroscopic systems, the fidelity-based counterpart is a
more suitable quantity to study. Finally, we apply the previous results to two
representative models of topological insulators in 1D and 2D.
