Abstract: This paper proposes a Ka-band coaxial multi-gap cavity operating in the TM₅₁-2π mode. The CST eigenmode solver is used to study the characteristics of the electric field distribution, and the mode characteristics of this cavity have been analyzed based on the all-pass coupling structure at the outer radius. By combining space-charge wave theory and 3-D particle-in-cell (PIC) simulation analysis, this paper studies the start-oscillation characteristics of the high-order mode coaxial multi-gap cavity using the multi-beam excitation method. And it analyzes the mode stability and beam-wave interaction characteristics of the coaxial multi-gap cavity operating in the high-order mode. The results show that the coaxial multi-gap cavity operating in the TM₅₁-2π mode adopting the coupling method at the outer radius possesses high mode stability. In this structure, multiple beams can not only uniformly inspire the operating mode but also non-uniformly inspire the competition mode. Different from the multi-beam extended-interaction klystron (EIK) operating in the fundamental mode, the high-order mode EIK with this structure establishes the gap voltages separately. Therefore, the peak electric fields with different phases can interact with the beams respectively. While keeping the same total beam current and beam voltage, the operating method driven by more beams requires a smaller focusing magnetic field.