  We study the thermodynamic properties of topological Josephson junctions
using a quantum spin Hall (QSH) insulator-based junction as an example. In
particular, we propose that phase-dependent measurements of the heat capacity
offer an alternative to Josephson-current measurements to demonstrate key
topological features. Even in an equilibrium situation, where the fermion
parity is not conserved, the heat capacity exhibits a pronounced double peak in
its phase dependence as a signature of the protected zero-energy crossing in
the Andreev spectrum. This double-peak feature is robust against changes of the
tunneling barrier and thus allows one to distinguish between topological and
trivial junctions. At short time scales fermion parity is conserved and the
heat capacity is $4\pi$-periodic in the superconducting phase difference. We
propose a dispersive setup coupling the Josephson junction to a tank LC circuit
to measure the heat capacity of the QSH-based Josephson junction sufficiently
fast to detect the $4\pi$-periodicity. Although explicitly calculated for a
short QSH-based Josephson junction, our results are also applicable to long as
well as nanowire-based topological Josephson junctions.
