  The goal of quantum metrology is the exploitation of quantum resources, like
entanglement or quantum coherence, in the fundamental task of parameter
estimation. Here we consider the question of the estimation of the Unruh
temperature in the scenario of relativistic quantum metrology. Specifically, we
study two distinct cases. First, a single Unruh-DeWitt detector interacting
with a scalar quantum field undergoes an uniform acceleration for a finite
amount of proper time, and the role of coherence in the estimation process is
analyzed. After this, we consider two initially entangled detectors, one of
which is inertial while the other one undergoes acceleration. Our results show
that the maximum of the Fisher information, thus characterizing the maximum
possible precision according to Cramm\'{e}r-Rao bound, occurs only for small
accelerations, while it decreases fast when acceleration increases. Moreover,
the role of initial coherence ---in the single detector case---, or
entanglement ---in the two detectors case---, is to decrease Fisher
information. Therefore, under the considered protocol, internal coherence (or
entanglement) is not a resource for estimating Unruh temperature. These
unexpected results show that a detection of the Unruh effect can be even more
challenge than previously thought. Finally, by considering the connection
between Unruh effect and Hawking radiation, we discuss how our results can be
understood in the context of the estimation of Hawking temperature.
