  There exist severe limitations on the accuracy of low-temperature
thermometry, which poses a major challenge for future quantum-technological
applications. Low-temperature sensitivity might be manipulated by tailoring the
interactions between probe and sample. Unfortunately, the tunability of these
interactions is usually very restricted. Here, we focus on a more practical
solution to boost thermometric precision -- driving the probe. Specifically, we
solve for the limit cycle of a periodically modulated linear probe in an
equilibrium sample. We treat the probe-sample interactions \textit{exactly} and
hence, our results are valid for arbitrarily low temperatures $ T $ and any
spectral density. We find that weak near-resonant modulation strongly enhances
the signal-to-noise ratio of low-temperature measurements, while causing
minimal back action on the sample. Furthermore, we show that near-resonant
driving changes the power law that governs thermal sensitivity over a broad
range of temperatures, thus `bending' the fundamental precision limits and
enabling more sensitive low-temperature thermometry. We then focus on a
concrete example -- impurity thermometry in an atomic condensate. We
demonstrate that periodic driving allows for a sensitivity improvement of
several orders of magnitude in sub-nanokelvin temperature estimates drawn from
the density profile of the impurity atoms. We thus provide a feasible upgrade
that can be easily integrated into low-$T$ thermometry experiments.
