Gas influx during managed pressure drilling alters the annular hydrostatic and frictional pressures as the gas migrates and expands, complicating bottom-hole pressure (BHP) regulation. Control-oriented two-phase-flow models that neglect dynamic gas temperature may inaccurately predict gas expansion and pressure response. This study develops a thermally coupled four-state model and a feedback-linearization-based model predictive control (FL-MPC) strategy for gas migration before the gas reaches the wellhead. The model incorporates the local annular-liquid temperature, interfacial heat transfer, pressure–volume work, and the adopted gas equation of state. The wellhead-pressure-to-BHP channel has a relative degree of two, while the feedback-linearization gain and coordinate transformation remain regular over the investigated trajectory. On the zero-output manifold at 508 bar, the gas-volume derivative remained positive at all 1201 sampled states, with a minimum internal growth coefficient of 2.71 × 10⁻⁴ s⁻¹. In a separate 2500-step OLGA validation case, thermal coupling reduced the gas-temperature RMSE from 6.354 to 0.730 K and the BHP RMSE from 2.199 to 2.091 bar. At the 508 bar setpoint, FL-MPC entered the settling band at step 104, compared with steps 123 and 162 for NMPC and PID, respectively, and achieved an RMSE of 0.814 bar. Its average computation time was 0.195 ms per step, compared with 22.65 ms for NMPC. Robustness tests under small parameter, initial-state, and measurement perturbations maintained BHP tracking and satisfied the actuator constraints.