We examine in detail the relation between the optical gain spectra, mode-locked optical emission spectra, and temporal optical pulse widths as a function of temperature between 80 and 300 K in passively mode-locked InAs quantum dot lasers. By increasing the length of the active region, we can decrease the threshold gain requirement for mode locking. At 300 K, where the dot states and wetting layer are close to thermal equilibrium, the bandwidth of the optical emission spectra and temporal optical pulse width remain largely unaffected when the threshold gain requirement is reduced. At 80 K, where the dots are randomly populated, there is a near doubling of the optical bandwidth for the same reduction of the threshold gain requirement and a corresponding decrease in the temporal optical pulse width. Rate equations, which take explicit account of the photon density in the cavity, are used to qualitatively highlight the key parameters, which are responsible for increasing the optical bandwidth in the random population regime.