  We consider the problem of estimating the temperature $ T $ of a very cold
equilibrium sample. The temperature estimates are drawn from measurements
performed on a quantum probe strongly coupled to it. We model this scenario by
resorting to the canonical Caldeira-Leggett Hamiltonian and find analytically
the exact stationary state of the probe for arbitrary coupling strength. In
general, the probe does not reach thermal equilibrium with the sample, due to
their non-perturbative interaction. We argue that this is advantageous for low
temperature thermometry, as we show in our model that: (i) The thermometric
precision at low $ T $ can be significantly enhanced by strengthening the
probe-sampling coupling, (ii) the variance of a suitable quadrature of our
Brownian thermometer can yield temperature estimates with nearly minimal
statistical uncertainty, and (iii) the spectral density of the probe-sample
coupling may be engineered to further improve thermometric performance. These
observations may find applications in practical nanoscale thermometry at low
temperatures---a regime which is particularly relevant to quantum technologies.
