A high-power X-band klystron employing a double-gap output cavity has been operating at SLAC. Multi-gap output circuits have lower surface gradients at the interaction gaps than single-gap ones but are prone to self-oscillate due to negative beam loading and trapped higher-order modes. In the double-gap circuit design, considerable attention had been directed to deal with these stability problems. The performance of the present tube appears to be limited by gap breakdown and beam interception particularly at long pulses. A three-gap output cavity is currently under development to further reduce the gap surface gradient. Another new feature of the circuit is an enlarged downstream drift tube to improve on beam clearance. This paper discusses the considerations involved in designing a multi-gap output cavity and presents the cold test measurements on the three-gap circuit. The experimental data is compared with numerical results from the 3-D simulation code ARGUS.