  Motivated by recent transport measurements on the candidate spin-liquid phase
of the organic triangular lattice insulator EtMe$_3$Sb[Pd(dmit)$_2$]$_2$, we
perform a controlled calculation of the thermal conductivity at intermediate
temperatures in a spin liquid system where a spinon Fermi surface is coupled to
a $U(1)$ gauge field. The present computation builds upon the double expansion
approach developed by Mross \emph{et al.} [Phys. Rev. B \textbf{82}, 045121
(2010)] for small $\epsilon=z_b -2$ (where $z_b$ is the dynamical critical
exponent of the gauge field) and large number of fermionic species $N$. Using
the so-called memory matrix formalism that most crucially does not assume the
existence of well-defined quasiparticles at low energies in the system, we
calculate the temperature dependence of the thermal conductivity $\kappa$ of
this model due to non-critical Umklapp scattering of the spinons for a finite
$N$ and small $\epsilon$. Then we discuss the physical implications of such
theoretical result in connection with the experimental data available in the
literature.
