Recently proposed neutrino emission processes in high-density matter r
esult in the relatively rapid cooling of a neutron star's interior fol
lowed by a precipitous drop in the surface temperature. We show that t
he time interval between the formation of the neutron star and the dro
p in the surface temperature is primarily determined by the structure
of the neutron star and is relatively insensitive to the rapid cooling
mechanism itself. Thus, observations of thermal emissions from neutro
n stars have the potential for constraining the neutron star's structu
re and the underlying equation of state of dense matter.