  We study the time evolution of thermodynamic observables that characterise
the dissipative nature of thermal relaxation after an instantaneous temperature
quench. Combining tools from stochastic thermodynamics and large-deviation
theory, we develop a powerful theory for computing the large-deviation
statistics of such observables. Our method naturally leads to a description in
terms of a dynamical Landau theory, a versatile tool for the analysis of
finite-time dynamical phase transitions. The topology of the associated Landau
potential allows for an unambiguous identification of the dynamical order
parameter and of the phase diagram. As an immediate application of our method,
we show that the probability distribution of the heat exchanged between a
mean-field spin model and the environment exhibits a singular point, a kink,
caused by a finite-time dynamical phase transition. Using our Landau theory, we
conduct a detailed study of the phase transition. Although the manifestation of
the new transition is similar to that of a previously found finite-time
transition in the magnetisation, the properties and the dynamical origins of
the two turn out to be very different.
