  Topological lasers are of growing interest as a way to achieve
disorder-robust single mode lasing using arrays of coupled resonators. We study
lasing in a two-dimensional coupled resonator lattice exhibiting transitions
between trivial and topological phases, which allows us to systematically
characterize the lasing modes throughout a topological phase. We show that,
unlike conventional topological robustness that requires a sufficiently large
bulk band gap, bifurcations in topological edge mode lasing can occur even when
the band gap is maximized. We show that linear mode bifurcations from single
mode to multi-mode lasing can occur deep within the topological phase,
sensitive to both the pump shape and lattice geometry. We suggest ways to
suppress these bifurcations and preserve single edge mode lasing.
