  Lasing in the bad cavity regime has promising applications in precision
metrology due to the reduced sensitivity to cavity noise. Here we investigate
the spectral properties and phase behavior of pulsed lasing on the $^1$S$_0 -
^3$P$_1$ line of $^{88}$Sr in a mK thermal ensemble, as first described in
arxiv:1903.12593. The system operates in a regime where the Doppler-broadened
atomic transition linewidth is several times larger than the cavity linewidth.
We find that by detuning the cavity resonance, the influence of the cavity
noise on the peak lasing frequency can be eliminated to first order despite the
system not being deep in the bad cavity regime. Experimental results are
compared to a model based on a Tavis-Cummings Hamiltonian, which enables us to
investigate the interplay between different thermal velocity classes as the
underlying mechanism for the reduction in cavity noise. These
velocity-dependent dynamics can occur in pulsed lasing and during the turn-on
behavior of lasers in the superradiant crossover regime.
