  There is increasing interest in discrete or "pixelated" spacetime models as a
foundation for a satisfactory theory of quantum gravity. If spacetime possesses
a cellular structure, there should be observable consequences: for example, the
vacuum becomes a dispersive medium. Of obvious interest are the implications
for the thermodynamic properties of quantum black holes. As a first step to
investigating that topic, we present here a calculation of the response of a
uniformly accelerating particle detector in the (modified) quantum vacuum of a
background pixelated spacetime, which is well known to mimic some features of
the Hawking effect. To investigate the detector response we use the standard
DeWitt treatment, with a two-point function modified to incorporate the
dispersion. We use dispersion relations taken from the so-called doubly special
relativity (DSR) and Ho\v{r}ava-Lifshitz gravity. We find that the correction
terms retain the Planckian nature of particle detection, but only for
propagation faster than the speed of light, a possibility that arises in this
treatment because the dispersion relations violate Lorentz invariance. A fully
Lorentz-invariant theory requires additional features; however, we believe the
thermal response will be preserved in the more elaborate treatment.
