  In comparison with conventional lasers, topological lasers are more robust
and can be immune to disorder or de-fects if lasing occurs in topologically
protected states. Previously reported topological lasers were almost
exclu-sively based on the first-order photonic topological insulators. Here, we
show that lasing can be achieved in the zero-dimensional corner state in a
second-order photonic topological insulator, which is based on Kagome
wave-guide array with a rhombic configuration. If gain is present in the corner
of the structure, where topological corner state resides, stable lasing in this
state is achieved, with lowest possible threshold, in the presence of uniform
loss-es and two-photon absorption. When gain acts in other corners of the
structure, lasing may occur in edge or bulk states, but it requires
substantially larger thresholds and transition to stable lasing occurs over
much larger propa-gation distances, sometimes due to instabilities, which are
absent for lasing in corner states. We find that increasing two-photon
absorption generally plays strong stabilizing action for nonlinear lasing
states. The transition to stable lasing stimulated by noisy inputs is
illustrated. Our work demonstrates the realistic setting for corner state laser
based on higher-order topological insulator realised with waveguide arrays.
