  In a Fermi superfluid increasing population imbalance leads initially to
reduction of the transition temperature, then the appearance of modulated
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, and finally the suppression of
pairing itself. For interaction strength such that the `balanced' system has a
normal state pseudogap, increasing imbalance reveals anomalous spectral
behavior. At a fixed weak imbalance (small polarization) the stable homogeneous
superfluid occurs only above a certain temperature. The density of states has a
minimum at the Fermi level, then a weak peak {\it within the gap}, and then the
large, gap edge, coherence features. On heating, this non monotonic energy
dependence changes to a more conventional fluctuation driven pseudogap, with a
monotonic energy dependence. At large imbalance the ground state is FFLO and
`pseudogapped' due to the modulated order. It changes to a gapless normal state
on heating, and then shows a pseudogap again at a higher temperature. These
weak imbalance and strong imbalance features both involve effects well beyond
mean field theory. We establish them by using a Monte Carlo technique on large
lattices, motivate the results in terms of the pairing field distribution, and
compare them to spectroscopic results in the imbalanced unitary Fermi gas.
