  We investigate the finite-temperature superfluid behavior of ultracold atomic
Fermi gases in quasi-two-dimensional Lieb lattices with a short-range
attractive interaction, using a pairing fluctuation theory within the BCS-BEC
crossover framework. We find that the presence of a flat band, along with van
Hove singularities, leads to exotic quantum phenomena. As the Fermi level
enters the flat band, both the gap and the superfluid transition temperature
$T_c$ as a function of interaction change from a conventional exponential
behavior into an unusual power law, and the evolution of superfluid densities
with temperature also follows a power law even at weak interactions. The
quantum geometric effects, manifested by an enhanced effective pair hopping
integral, may contribute significantly to both $T_c$ and the superfluidities.
As the chemical potential crosses the van Hove singularities in the weak
interaction regime, the nature of pairing changes between particle-like and
hole-like. A pair density wave state emerges at high densities with a
relatively strong interaction strength.
