  Temperature fluctuations are expected to be one of the limiting factors for
gravitational wave detectors in the very low frequency range. Here we report
the characterisation of this noise source in the LISA Pathfinder optical bench
and propose a method to remove its contribution from the data. Our results show
that temperature fluctuations are indeed limiting our measurement below one
millihertz, and that their subtraction leads to a factor 5.6 (15 dB) reduction
in the noise level at the lower end of the LISA measurement band 10^{-4} Hz,
which increases to 20.2 (26 dB) at even lower frequencies, i.e., 1.5x10^{-5}
Hz. The method presented here can be applied to the subtraction of other noise
sources in gravitational wave detectors in the general situation where multiple
sensors are used to characterise the noise source.
