  We propose an operational definition for the local temperature of a quantum
field employing Unruh-DeWitt detectors, as used in the study of the Unruh and
Hawking effects. With this definition, an inhomogeneous quantum system in
equilibrium can have different local temperatures, in analogy with the
Tolman-Ehrenfest theorem from general relativity. We study the local
temperature distribution on the ground state of hopping fermionic systems on a
curved background. The observed temperature tends to zero as the
thermometer-system coupling $g$ vanishes. Yet, for small but finite values of
$g$, we show that the product of the observed local temperature and the
logarithm of the local speed of light is approximately constant. Our
predictions should be testable on ultracold atomic systems.
