  We demonstrate that phase-dependent heat currents through
superconductor-topological insulator Josephson junctions provide a useful tool
to probe the existence of topological Andreev bound states, even for
multi-channel surface states. We predict that in the tunneling regime
topological Andreev bound states lead to a minimum of the thermal conductance
for a phase difference $\phi=\pi$, in clear contrast to a maximum of the
thermal conductance at $\phi=\pi$ that occurs for trivial Andreev bound states
in superconductor-normal metal tunnel junctions. This opens up the possibility
that phase-dependent heat transport can distinguish between topologically
trivial and nontrivial 4$\pi$ modes. Furthermore, we propose a superconducting
quantum interference device geometry where phase-dependent heat currents can be
measured using available experimental technology.
