  In the framework of open quantum systems, we study the geometric phase
acquired by freely falling and static two-level atoms interacting with
quantized conformally coupled massless scalar fields in de Sitter-invariant
vacuum. We find that, for the freely falling atom, the geometric phase gets a
correction resulting from a thermal bath with the Gibbons-Hawking temperature,
thus it clearly reveals the intrinsic thermal nature of de Sitter spacetime
from a different physical context. For the static atom, there is a correction
to the geometric phase coming from both the intrinsic thermal nature of de
Sitter spacetime and the Unruh effect associated with the proper acceleration
of the atom. Furthermore, in a gedanken experiment, we estimate the magnitude
of the correction to the geometric phase as opposed to that in a flat
spacetime. We find that the correction for the freely falling atom is too tiny
to be measured, and that for the static atom achieves an observable magnitude
only when the atom almost locates at the horizon.
