  We investigate superfluidity of bosons in gapped topological bands and
discover a new phase that has no counterparts in the previous literature. This
phase is characterized by a highly unconventional modulation of the order
parameter, breaking the crystallographic symmetry, and for which the
condensation momentum is neither zero nor any other high-symmetry vector of the
Brillouin zone. This unconventional structure impacts the spectrum of
Bogoliubov excitations and, consequently, the speed of sound in the system.
Even in the case of perfectly flat bands, the speed of sound and Bogoliubov
excitations remain nonvanishing, provided that the underlying topology and
quantum geometry are nontrivial. Furthermore, we derive detailed expressions
for the superfluid weight using the Popov hydrodynamic formalism for
superfluidity and provide estimates for the Berezinskii-Kosterlitz-Thouless
temperature, which is significantly enhanced by the nontriviality of the
underlying quantum metric. These results are applicable to generic topological
bosonic bands, with or without dispersion. To illustrate our findings, we
employ the Haldane model with a tunable bandwidth, including the narrow
lowest-band case. Within this model, we also observe a re-entrant superfluid
behavior: As the Haldane's magnetic flux is varied, the
Berezinskii-Kosterlitz-Thouless transition temperature initially decreases to
almost zero, only to resurface with renewed vigor.
