  We systematically investigate the ground state phase diagram and the finite
temperature phase transitions for a Rydberg-dressed Fermi gas loaded in a
bilayer optical lattice. When an effective finite-ranged attraction is induced,
our self-consistent mean-field calculation shows that the gapped topological (
$p$-wave) superfluids in each layer are coupled together by the $s$-wave
pairing in an intermediate inter-layer distance with a spontaneously modulated
phases between these two order parameters. The obtained ground state is a
gapless topological superfluid with quantized topological charges
characterizing the gapless points, leading to a zero energy flat band at the
edges. Finally, we calculate the finite temperature phase diagrams of this
two-dimensional gapless superfluid and observe two distinct critical
temperatures, demonstrating the fruitful many-body effects on a paired
topological superfluids.
