  We discuss the cooling history of the low-mass, helium core white dwarfs in
short orbital period millisecond pulsars PSR J0751+1807 and PSR J1012+5307. The
revised cooling age estimated by Alberts et al. agrees with the age estimation
for PSR J1012+5307, removing the discrepancy between the spin-down age and the
cooling age. However, if we accept this model then the helium white dwarf in
the binary pulsar system PSR J0751+1807 must be much hotter than is observed.
We propose that this discrepancy may be resolved if, after detachment of the
secondary star from its Roche lobe in PSR J0751+1807, the star loses its
hydrogen envelope due to pulsar irradiation. When hydrogen burning stops, the
white dwarf will cool down much more quickly than in the case of a thick
hydrogen envelope with a hydrogen burning shell. We discuss several
possibilities to explain different cooling histories of white dwarfs in both
systems.
