As one approaches the absolute zero, the noise-to-signal ratio of temperature estimates diverges.
Hence, it does matter, how fast is this precision loss.
At best, the relative error grows as a power law as T → 0. And the exponent is set by the low-frequency dependence of the density of states of the sample being measured.
So, if the precision scaling depends on the sample we are given, there is little that we can do, right?
Here is the thing: We can still modulate the potential that confines the probe we are using to measure the sample. Done right, this can "BEND" the precision scaling law.
We demonstrate this with a Brownian particle in a linear bath. The calculation is painful and long, and FUN! All details are in our generous appendices!
But wait! Is it a good idea to take a cold sample, couple it to a probe, and then keep pumping energy by driving it?
That sounds like a recipe for disaster...
Only it isn't. We show that the steady-state heating of the sample is negligible.