Abstract: To achieve GW-level amplification output radiation at X-band, a relativistic triaxial klystron amplifier （TKA） with two-stage cascaded double-gap bunching cavities is investigated. The input cavity is optimized to obtain a high absorption rate of the external injection microwave. The cascaded bunching cavities are optimized to achieve a high depth of the fundamental harmonic current. A double-gap standing wave extractor is designed to improve the beam wave conversion efficiency. Two reflectors with high reflection coefficients both to the asymmetric mode and the TEM mode are employed to suppress the asymmetric mode competition and TEM mode microwave leakage. Particle-in-cell simulation results show that a high power microwave with a power of 2.53 GW and a frequency of 8.4 GHz is generated with a 690 kV, 9.3 kA electron beam excitation and a 25 kW radio-frequency signal injection. Meanwhile, there is insignificant self-excitation of parasitic mode in the proposed structure by adopting the reflectors. The relative phase difference between the injected signals and the output microwaves keeps locked after the amplifier became saturated.