The knowledge of the thermal radiative and thermodynamic properties of uranium and plutonium carbides under extreme conditions is essential for designing new metallic fuel materials for next generation of the nuclear reactors. The present work is devoted to the study of the thermal radiative and thermodynamic properties of the liquid and solid uranium and plutonium carbides at their melting/freezing temperatures. The Stefan-Boltzmann law, total energy density, number density of photons, Helmholtz free energy density, internal energy density, enthalpy density, entropy density, heat capacity at constant volume, pressure, and the normal total emissivity are calculated using the experimental data for the frequency dependence of the normal spectral emissivity of liquid and solid uranium and plutonium carbides in the visible-near-infrared range. It is shown that the thermal radiative and thermodynamic functions of uranium carbide have a slight difference during the liquid-to-solid transition. Unlike UC, such a difference between these functions has not been established for the plutonium carbide. The calculated values for the normal total emissivity of the uranium and plutonium carbides at their melting temperatures are in good agreement with the experimental data. Based on a model of Hagen-Rubens and the Wiedemann-Franz law, a new noncontact optical method to determine the thermal conductivity of metals and carbides at their melting/freezing points is proposed.