  How long does a uniformly accelerated observer need to interact with a
quantum field in order to record thermality in the Unruh temperature? We
address this question for a pointlike Unruh-DeWitt detector, coupled linearly
to a real Klein-Gordon field of mass $m\ge0$ and treated within first order
perturbation theory, in the limit of large detector energy gap
$E_{\text{gap}}$. We first show that when the interaction duration $\Delta T$
is fixed, thermality in the sense of detailed balance cannot hold as
$E_{\text{gap}}\to\infty$, and this property generalises from the Unruh effect
to any Kubo-Martin-Schwinger state satisfying certain technical conditions. We
then specialise to a massless field in four spacetime dimensions and show that
detailed balance does hold when $\Delta T$ grows as a power-law in
$E_{\text{gap}}$ as $E_{\text{gap}}\to\infty$, provided the switch-on and
switch-off intervals are stretched proportionally to $\Delta T$ and the
switching function has sufficiently strong Fourier decay. By contrast, if
$\Delta T$ grows by stretching a plateau in which the interaction remains at
constant strength but keeping the duration of the switch-on and switch-off
intervals fixed, detailed balance at $E_{\text{gap}}\to\infty$ requires $\Delta
T$ to grow faster than any polynomial in $E_{\text{gap}}$, under mild technical
conditions. These results also hold for a static detector in a Minkowski heat
bath.
  The results limit the utility of the large $E_{\text{gap}}$ regime as a probe
of thermality in time-dependent versions of the Hawking and Unruh effects, such
as an observer falling into a radiating black hole. They may also have
implications on the design of prospective experimental tests of the Unruh
effect.
