  We analyse the response and thermal behaviour of an Unruh-DeWitt detector as
it travels through cosmological spacetimes, with special reference to the
question of how to define surface gravity and temperature in dynamical
spacetimes. Working within the quantum field theory on curved spacetime
approximation, we consider a detector as it travels along geodesic and
accelerated Kodama trajectories in de Sitter and asymptotically de Sitter FLRW
spacetimes. By modelling the temperature of the detector using the
detailed-balance form of the Kubo--Martin--Schwinger (KMS) conditions as it
thermalises, we can better understand the thermal behaviour of the detector as
it interacts with the quantum field, and use this to compare competing
definitions of surface gravity and temperature that persist in the literature.
These include the approaches of Hayward-Kodama, Ashtekar et al., Fodor et al.,
and Nielsen-Visser. While these are most often examined within the context of a
dynamical black hole, here we shift focus to surface gravity on the evolving
cosmological horizon.
