  Recent study on doping effects in the heavy fermion superconductor CeCoIn$_5$
has shown that a small amount of doping induces unexpectedly large broadening
of the second order transition into the high field and low temperature (HFLT)
phase of this material. Motivated by this observation, effects of quenched
disorder on the second order transition into a longitudinal
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state are examined. The observed large
broadening of the transition is naturally explained as a consequence of
softness of each FFLO nodal plane. The present results strongly support the
scenario identifying the HFLT phase of CeCoIn$_5$ with a longitudinal FFLO
vortex state.
