  We develop a general perturbative theory of finite-coupling quantum
thermometry up to second order in probe-sample interaction. By assumption, the
probe and sample are in thermal equilibrium, so the probe is described by the
mean-force Gibbs state. We prove that the ultimate thermometric precision can
be achieved - to second order in the coupling - solely by means of local energy
measurements on the probe. Hence, seeking to extract temperature information
from coherences or devising adaptive schemes confers no practical advantage in
this regime. Additionally, we provide a closed-form expression for the quantum
Fisher information, which captures the probe's sensitivity to temperature
variations. Finally, we benchmark and illustrate the ease of use of our
formulas with two simple examples. Our formalism makes no assumptions about
separation of dynamical timescales or the nature of either the probe or the
sample. Therefore, by providing analytical insight into both the thermal
sensitivity and the optimal measurement for achieving it, our results pave the
way for quantum thermometry in setups where finite-coupling effects cannot be
ignored.
