  We consider p-wave (triplet) pairing of single hyperfine state ultracold
atomic gases trapped in quasi-two-dimensional optical lattices. We find that
the critical temperatures in the lattice model is considerably higher and
experimentally attainable around half-filling in contrast to the predictions of
continuum model for p-wave superfluids. In tetragonal lattices, we show that
the atomic compressibility and spin susceptibility have a peak at low
temperatures exactly at the half-filling, but this peak splits into two in the
orthorhombic lattices. These peaks reflect the p-wave structure of the order
parameter for superfluidity and they disappear as the critical temperature is
approached from below. We also calculate the superfluid density tensor, and
show that for the orthorhombic case there is no off-diagonal component, however
in the tetragonal case an off-diagonal component develops, and becomes a key
signature of the exotic p-wave state.
