Developing semiconducting materials with suitable band gap has attracted increasing attention for next-generation intelligent electronic devices. Recently, the newly synthesized monolayer carbon boron (C3B) material exhibits outstanding electronic properties and processes with indirect bandgap, which is considered as the promising alternatives of graphene. The heat conduction capability plays a critical role in the performance of C3B-based electronic devices and thermocatalysis applications. Herein, the thermal conductivity of monolayer C3B and graphene, and interfacial phonon transport across heterointerface are systematically investigated utilizing non-equilibrium molecular dynamics simulation. Compared with graphene, the reduced thermal conductivity of C3B sheet mainly stems from the decreased phonon group velocity and phonon relaxation time when the periodical boron atoms are introduced in C3B sheet based on the lattice dynamics and spectral phonon transmission analysis. Meanwhile, the influences of temperature and strain on the thermal conductivity of C3B are also discussed. Moreover, the thermal transport across the graphene|C3B heterointerface with different temperatures is analyzed. Interestingly, it is found that the larger interfacial thermal conductance of zigzag heterointerfaces mainly originates from the stronger phonon-phonon coupling between the out-of-plane vibrational modes of C3B and graphene. Our work would provide more insights for fundamental understanding the thermal transport properties of two-dimensional materials and be beneficial to the design of thermal management and thermoelectric materials.