Pseudospark-sourced electron beam is a promising candidate for driving vacuum electronic devices to generate millimeter wave and terahertz wave radiation as it has a very high combined beam current density. However, the inherent velocity spread of the beam, which is difficult to measure in experiment, has a great influence on the operating frequency and efficiency of the vacuum electronic device. In this paper, the velocity distribution characteristics of the electron beam produced by a single-gap hollow cathode electron gun are numerically studied and a three-dimensional kinetic plasma simulation model of a single-gap hollow cathode electron gun is built by using particle in cell and Monte Carlo collision methods in Vorpal. Based on the simulation model, the time-dependent evolution of the plasma formation inside the hollow cathode and electron beam generation process are observed. It is demonstrated that the pseudospark-sourced electron beam has a relatively large velocity spread. The time-dependent velocity distribution of the beam is analyzed, and the dependence of the beam velocity distribution under various operating conditions such as anode–cathode potential difference, gas pressure, and cathode aperture size are also studied.