  We study the problem of estimating the temperature of Gaussian systems with
feasible measurements, namely Gaussian and photo-detection-like measurements.
For Gaussian measurements, we develop a general method to identify the optimal
measurement numerically, and derive the analytical solutions in some relevant
cases. For a class of single-mode states that includes thermal ones, the
optimal Gaussian measurement is either Heterodyne or Homodyne, depending on the
temperature regime. This is in contrast to the general setting, in which a
projective measurement in the eigenbasis of the Hamiltonian is optimal
regardless of temperature. In the general multi-mode case, and unlike the
general unrestricted scenario where joint measurements are not helpful for
thermometry (nor for any parameter estimation task), it is open whether joint
Gaussian measurements provide an advantage over local ones. We conjecture that
they are not useful for thermal systems, supported by partial analytical and
numerical evidence. We further show that Gaussian measurements become optimal
in the limit of large temperatures, while on/off photo-detection-like
measurements do it for when the temperature tends to zero. Our results
therefore pave the way for effective thermometry of Gaussian quantum systems
using experimentally realizable measurements.
