  We investigate exotic paired states of spin-imbalanced Fermi gases in
anisotropic lattices, tuning the dimension between one and three. We calculate
the finite temperature phase diagram of the system using real-space dynamical
mean-field theory in combination with the quantum Monte Carlo method. We find
that regardless of the intermediate dimensions examined, the
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state survives to reach about one third
of the BCS critical temperature of the spin-density balanced case. We show how
the gapless nature of the state found is reflected in the local spectral
function. While the FFLO state is found at a wide range of polarizations at low
temperatures across the dimensional crossover, with increasing temperature we
find out strongly dimensionality-dependent melting characteristics of shell
structures related to harmonic confinement. Moreover, we show that intermediate
dimension can help to stabilize an extremely uniform finite temperature FFLO
state despite the presence of harmonic confinement.
