The formation and development of pseudospark discharge, especially the onset of the breakdown, are of great technological interests in multiple applications due to their influences on the limits of current rising and fast switching performances of the devices. In this work, the development of pseudospark discharge in the pre-discharge and hollow cathode phases in a single-gap device are investigated by a time-dependent model to calculate the temporal development of total ionization cross section in varying times and regions. The simulations in our work are performed using the two-dimensional kinetic plasma simulation code XOOPIC. The time-dependent evolutions of the ionization cross section in pre-discharge and hollow cathode phases are presented under varying electric fields and hollow cathode configurations. Thus the electron multiplications and plasma generation processes by ionizing collisions in varying phases are examined and their dependences on a variety of external parameters are determined in different regions in the pseudospark device. A sequence of physical events and their influences in different regions are also identified via the quantitative analysis of time-dependent ionization cross section. The discharge formation time shows highest dependences on the cathode aperture diameters and anode voltages. Additionally, a linear dependence of the pseudospark breakdown time on the time-averaged ionization cross section is illustrated under varying external parameters. It indicates that the influences of the external parameters on the discharge performances can be determined and estimated via the total and average ionization cross sections under varying external conditions. In this work, both a qualitative understanding of the pseudospark onset mechanism and a quantitative approach to estimate the formation time in a pseudospark device with varying parameters are developed via this model.