  Assuming that two-component Fermi gases with opposite artificial magnetic
fields on a square optical lattice are well-described by the so-called
time-reversal-symmetric Hofstadter-Hubbard model, we explore the thermal
superfluid properties along with the critical Berezinskii-Kosterlitz-Thouless
(BKT) transition temperature in this model over a wide-range of its parameters.
In particular, since our self-consistent BCS-BKT approach takes the multi-band
butterfly spectrum explicitly into account, it unveils how dramatically the
inter-band contribution to the phase stiffness dominates the intra-band one
with an increasing interaction strength for any given magnetic flux.
