  The problem of estimating the frequency of a two-level atom in a noisy
environment is studied. Our interest is to minimise both the energetic cost of
the protocol and the statistical uncertainty of the estimate. In particular, we
prepare a probe in a "GHZ-diagonal" state by means of a sequence of qubit gates
applied on an ensemble of $ n $ atoms in thermal equilibrium. Noise is
introduced via a phenomenological time-nonlocal quantum master equation, which
gives rise to a phase-covariant dissipative dynamics. After an interval of free
evolution, the $ n $-atom probe is globally measured at an interrogation time
chosen to minimise the error bars of the final estimate. We model explicitly a
measurement scheme which becomes optimal in a suitable parameter range, and are
thus able to calculate the total energetic expenditure of the protocol.
Interestingly, we observe that scaling up our multipartite entangled probes
offers no precision enhancement when the total available energy $ \mathcal{E} $
is limited. This is at stark contrast with standard frequency estimation, where
larger probes---more sensitive but also more "expensive" to prepare---are
always preferred. Replacing $ \mathcal{E} $ by the resource that places the
most stringent limitation on each specific experimental setup, would thus help
to formulate more realistic metrological prescriptions.
