  Neutron stars cooling passively since their birth may be reheated in their
late-stage evolution by a number of possible phenomena: rotochemical, vortex
creep, crust cracking, magnetic field decay, or more exotic processes such as
removal of neutrons from their Fermi seas (the nucleon Auger effect), baryon
number-violating nucleon decay, and accretion of particle dark matter. Using
Exposure Time Calculator tools, we show that reheating mechanisms imparting
effective temperatures of 2000--40000 Kelvin may be uncovered with excellent
sensitivities at the James Webb Space Telescope (JWST), the Extremely Large
Telescope (ELT), and the Thirty Meter Telescope (TMT), with imaging instruments
operating from visible-edge to near-infrared. With a day of exposure, they
could constrain the reheating luminosity of a neutron star up to a distance of
500 pc, within which about $10^5$ (undiscovered) neutron stars lie. Detection
in multiple filters could overconstrain a neutron star's surface temperature,
distance from Earth, mass, and radius. Using publicly available catalogues of
newly discovered pulsars at the FAST and CHIME radio telescopes and the
Galactic electron distribution models YMW16 and NE2001, we estimate the
pulsars' dispersion measure distance from Earth, and find that potentially
30$-$40 of these may be inspected for late-stage reheating within viable
exposure times, in addition to a few hundred candidates already present in the
ATNF catalogue. Whereas the coldest neutron star observed (PSR J2144$-$3933)
has an upper limit on its effective temperature of about 33000 Kelvin with the
Hubble Space Telescope, we show that the effective temperature may be
constrained down to 20000 Kelvin with JWST-NIRCam, 15000 Kelvin at ELT-MICADO,
and 9000 Kelvin with TMT-IRIS. Campaigns to measure thermal luminosities of old
neutron stars would be transformative for astrophysics and fundamental physics.
