  The recently discovered dynamical phase transition denotes non-analytic
behavior in the real time evolution of quantum systems in the thermodynamic
limit and has been shown to occur in different systems at zero temperature
[Heyl et al., Phys. Rev. Lett. 110, 135704 (2013)]. In this paper we extend the
analysis to non-zero temperature by studying a generalized form of the
Loschmidt echo, the work distribution function, of a quantum quench in the
transverse field Ising model. Although the quantitative behavior at non-zero
temperatures still displays features derived from the zero temperature
non-analyticities, it is shown that in this model dynamical phase transitions
do not exist if $T>0$. This is a consequence of the system being initialized in
a thermal state. Moreover, we elucidate how the Tasaki-Crooks-Jarzynski
relation can be exploited as a symmetry relation for a global quench or to
obtain the change of the equilibrium free energy density.
