  We theoretically investigate an ultracold spin-polarized atomic Fermi gas
with resonant odd-channel ($p$-wave) interactions trapped in one-dimensional
harmonic traps. We solve the Yang-Yang thermodynamic equations based on the
exact Bethe ansatz solution, and predict the finite-temperature density profile
and breathing mode frequency, by using a local density approximation to take
into account the harmonic trapping potential. The system features an exotic
super fermionic Tonks-Girardeau (super-fTG) phase, due to the large effective
range of the interatomic interactions. We explore the parameter space for such
a fascinating super-fTG phase at finite temperature and provide smoking-gun
signatures of its existence in both breathing mode frequencies and density
profiles. Our results suggest that the super-fTG phase can be readily probed at
temperature at about $0.1T_{F}$, where $T_{F}$ is the Fermi temperature. These
results are to be confronted with future cold-atom experiments with $^{6}$Li
and $^{40}$K atoms.
