  We present a study of longitudinal thermal transport in the Kitaev spin model
on the honeycomb lattice, focusing on the role of anisotropic exchange to cover
both, gapless and gapped phases. Employing a complementary combination of exact
diagonalization on small systems and an average gauge configuration approach
for up to $\sim O(10^4)$ spinful sites, we report results for the dynamical
energy current auto-correlation function as well as the dc thermal conductivity
over a wide range of temperatures and exchange anisotropies. Despite a
pseudogap in the current correlation spectra, induced by emergent thermal gauge
disorder on any finite system, we find that in the thermodynamic limit, gapless
and gapful phases both feature normal dissipative transport, with a temperature
dependence crossing over from power law to exponentially activated behavior
upon gap opening.
