  We present a theory for quasiparticle heat transport through superconducting
weak links. The thermal conductance depends on the phase difference ($\phi$) of
the superconducting leads. Branch conversion processes, low-energy Andreev
bound states near the contact and the suppression of the local density of
states near the gap edge are related to phase-sensitive transport processes.
Theoretical results for the influence of junction transparency, temperature and
disorder, on the phase modulation of the conductance are reported. For
high-transmission weak links, $D\to 1$, the formation of an Andreev bound state
at $\epsilon_{\text{\tiny b}}=\Delta\cos(\phi/2)$ leads to suppression of the
density of states for the continuum excitations that transport heat, and thus,
to a reduction in the conductance for $\phi\simeq\pi$. For low-transmission
($D\ll 1$) barriers resonant scattering at energies
$\epsilon\simeq(1+D/2)\Delta$ leads to an increase in the thermal conductance
as $T$ drops below $T_c$ (for phase differences near $\phi=\pi$).
