  Current phonon transport theory based on ground-state calculations has been
successful in predicting thermal conductivity at room and medium temperatures
but may misrepresent behavior at high temperatures. In this work, we predict
the thermal conductivity ($\kappa$) of ZrC including electronic and phonon
contributions from 300 K to 3500 K, by including high-order phonon scattering,
lattice expansion, temperature-dependent (TD) harmonic and anharmonic force
constants, and inter-band phonon conduction by using first principles. For the
phonon transport, we find that four-phonon scattering significantly reduces the
phonon thermal conductivity ($\kappa_{ph}$), e.g., by $\sim$60% and $\sim$75%
at 2500 K and 3500 K, respectively. After including four-phonon scattering and
all other factors, $\kappa_{ph}$ shows a $\sim$T$^{-1.5}$ rather than
$\sim$T$^{-1}$ dependence. The contribution from inter-band (Wigner) phonon
conduction is small, even at ultra-high temperatures. The temperature
dependence of anharmonic force constants decreases the phonon scattering
cross-section at elevated temperatures and increases the $\kappa_{ph}$
significantly (by 52% at 3500 K). For the electronic thermal transport, we find
that it is sensitive to and can be changed by 20% by the TD lattice constants.
The Lorenz number varies from 1.6 to 3.3$\times$10$^{-8}$
W$\cdot\Omega\cdot$K$^{-2}$ at different temperatures. The theoretical
prediction in the literature overpredicts $\kappa_{ph}$ (e.g., $\sim$28%) and
underpredicts the $\kappa_{el}$ (e.g., $\sim$38%), resulting in an overall
underprediction of $\kappa$ ($\sim$26% at 1500 K). The impacts of grain size
and defects are found strong, and no reported experimental data has reached the
intrinsic theoretical thermal conductivity of ZrC yet.
