  The Hofstadter energy spectrum of twisted bilayer graphene is found to have
recursive higher-order topological properties. We demonstrate that higher-order
topological insulator (HOTI) phases, characterized by localized corner states,
occur as replicas of the original HOTIs to fulfill the self-similarity of the
Hofstadter spectrum. We show the existence of the exact flux translational
symmetry of twisted bilayer graphene at all commensurate angles. Based on this
result, we carefully identify that the original HOTI phase at zero flux is
re-entrant at a half-flux periodicity, where the effective twofold rotation is
preserved. In addition, numerous replicas of the original HOTIs are found for
fluxes without protecting symmetries. Similar to the original HOTIs, replica
HOTIs feature both localized corner states and edge-localized real-space
topological markers. The replica HOTIs originate from the different interaction
scales, namely, intralayer and interlayer couplings, in twisted bilayer
graphene. The topological aspect of Hofstadter butterflies revealed in our
results highlights symmetry-protected topology in quantum fractals.
