In this paper, direct numerical simulations have been performed to explore the equivalence of different thermal boundary conditions in compressible turbulent channel flows at fixed Re=6000,Ma=1.5. Three canonical types of thermal boundary conditions will be investigated at almost equivalent setups, including the first boundary condition with fixed wall temperature T_{w} (constant Dirichlet boundary condition), the second boundary condition with fixed wall heat-flux q_{w} (constant Neumann boundary condition), and the third boundary condition (Robin boundary condition). The turbulent statistics of the temperature and velocity fields, including mean profiles, root-mean-square values, second-order statistics, and normalized probability density functions, temperature stripes near the wall and the budget of internal energy have been analyzed in detail to clarify the differences caused by the different thermal boundary conditions. The results show that the three thermal boundary conditions have almost negligible effect on the velocity field, whereas some discernible deviations can be observed for the temperature field in the near-wall region with y^{+}≲30. Furthermore, the statistics from the second and third thermal boundary conditions are very close, enabling the usage of the second boundary condition to mimic the more complex third boundary conditions.