We develop a self-consistent theoretical model for simulating the lasing characteristics of photonic-crystal surface-emitting lasers (PCSELs) under continuous-wave (CW) operation that takes into account thermal effects caused by current injection. Our model enables us to analyze the lasing characteristics of PCSELs under CW operation by solving self-consistently the changes in the in-plane optical gain and refractive index distribution, which is associated with heat generation and temperature rise, and the change in the oscillation modes. We reveal that the lasing band-edge selectivity and beam quality of the PCSELs are affected by the spatial distribution of the band-edge frequency of the photonic crystal formed by the refractive index distribution, which depends on the temperature distribution in the resonator. Furthermore, we show that single-mode lasing with narrow beam divergence can be realized even at high current injection under CW operation by introducing a photonic-crystal structure with an artificially formed lattice constant distribution, which compensates such band-edge frequency distribution.