Type Iax supernovae may arise from failed explosions of white dwarfs that leave behind a bound remnant (i.e., a postgenitor star) that could be identified in wide field surveys. To understand their observational signatures, we simulate these white dwarf (WD) postgenitors from shortly after explosion until they move back down the WD cooling track, and we consider several possible WD masses and explosion energies. To predict the peculiar surface abundances of the WD postgenitors, our models take into account gravitational settling and radiative levitation. We find that radiative levitation is significant at temperatures above a mass-dependent critical temperature, typically in the range Teff ~ 50-100 * 10^3 K, significantly increasing surface abundances of iron-group elements. Due to enhanced iron group opacity compared to normal WDs, the postgenitor peak luminosity and cooling timescale depend sensitively on mass, with more massive WDs becoming brighter but cooling much faster. We discuss our results in light of recently discovered hypervelocity white dwarfs with peculiar surface compositions, finding that our low-mass postgenitor models match many of their observational characteristics. Finally, we explore the effects of thermohaline diffusion, tentatively finding that it strongly suppresses abundance enhancements created by radiative levitation, but more realistic modeling is required to reach a firm conclusion.
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