  One of the main advantages expected from using quantum probes as thermometers
is non invasiveness, i.e., a negligible perturbation to the thermal sample.
However, invasiveness is rarely investigated explicitly. Here, focusing on a
pure-dephasing spin probe in a bosonic sample, we show that there is a
non-trivial relation between the information on the temperature gained by a
quantum probe and the heat absorbed by the sample due to the interaction. We
show that optimizing over the probing time, i.e. considering a time-optimal
probing scheme, also has the benefit of limiting the heat absorbed by the
sample in each shot of the experiment. For such time-optimal protocols, we show
that it is advantageous to have very strong probe-sample coupling, since in
this regime the accuracy increases linearly with the coupling strength, while
the amount of heat per shot saturates to a finite value. Since in
pure-dephasing models the absorbed heat corresponds to the external work needed
to couple and decouple the probe and the sample, our results also represent a
first step towards the analysis of the thermodynamic and energetic cost of
quantum thermometry.
