  Relativistic quantum metrology studies the maximal achievable precision for
estimating a physical quantity when both quantum and relativistic effects are
taken into account. We study the relativistic quantum metrology of temperature
in (3+1)-dimensional de Sitter and anti-de Sitter space. Using Unruh-DeWitt
detectors coupled to a massless scalar field as probes and treating them as
open quantum systems, we compute the Fisher information for estimating
temperature. We investigate the effect of acceleration in dS, and the effect of
boundary condition in AdS. We find that the phenomenology of the Fisher
information in the two spacetimes can be unified, and analyze its dependence on
temperature, detector energy gap, curvature, interaction time, and detector
initial state. We then identify estimation strategies that maximize the Fisher
information and therefore the precision of estimation.
