  Recent advances in manipulation of orbital angular momentum (OAM) within the
paradigm of orbitronics present a promising avenue for the design of future
electronic devices. In this context, the recently observed orbital Hall effect
(OHE) occupies a special place. Here, focusing on both the second-order
topological and quantum anomalous Hall insulators in two-dimensional
ferromagnets, we demonstrate that topological phase transitions present an
efficient and straightforward way to engineer the OHE, where the OAM
distribution can be controlled by the nature of the band inversion. Using
first-principles calculations, we identify Janus RuBrCl and three septuple
layers of MnBi$_2$Te$_4$ as experimentally feasible examples of the proposed
mechanism of OHE engineering by topology. With our work we open up new
possibilities for innovative applications in topological spintronics and
orbitronics.
