  We construct a practical method for finding optimal Gaussian probe states for
the estimation of parameters encoded by Gaussian unitary channels. This method
can be used for finding all optimal probe states, rather than focusing on the
performance of specific states as shown in previous studies. As an example, we
apply this method to find optimal probes for the channel that combines the
phase-change and squeezing channels, and for generalized two-mode squeezing and
mode-mixing channels. The method enables a comprehensive study of temperature
effects in Gaussian parameter estimation. It has been shown that the precision
in parameter estimation using single mode states can be enhanced by increasing
the temperature of the probe. We show that not only higher temperature, but
also larger temperature differences between modes of a Gaussian probe state can
enhance the estimation precision.
