  We investigate the finite temperature properties of an ultracold atomic Fermi
gas with spin population imbalance in a highly elongated harmonic trap.
Previous studies at zero temperature showed that the gas stays in an exotic
spatially inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid
state at the trap center; while moving to the edge, the system changes into
either a non-polarized Bardeen-Cooper-Schriffer superfluid ($P<P_c$) or a fully
polarized normal gas ($P>P_c$), depending on the smallness of the spin
polarization $P$, relative to a critical value $P_c$. In this work, we show how
these two phase-separation phases evolve with increasing temperature, and
thereby construct a finite temperature phase diagram. For typical interactions,
we find that the exotic FFLO phase survives below one-tenth of Fermi degeneracy
temperature, which seems to be accessible in the current experiment. The
density profile, equation of state, and specific heat of the polarized system
have been calculated and discussed in detail. Our results are useful for the
on-going experiment at Rice University on the search for FFLO states in
quasi-one-dimensional polarized Fermi gases.
