  The ultimate precision of any measurement of the temperature of a quantum
system is the inverse of the local quantum thermal susceptibility [De Pasquale
et al., Nature Communications 7, 12782 (2016)] of the subsystem with whom the
thermometer interacts. If this subsystem can be described with the canonical
ensemble, such quantity reduces to the variance of the local Hamiltonian, that
is proportional to the heat capacity of the subsystem. However, the canonical
ensemble might not apply in the presence of interactions between the subsystem
and the rest of the system. In this work we address this problem in the
framework of locally interacting quantum systems. We prove that the local
quantum thermal susceptibility of any subsystem is close to the variance of its
local Hamiltonian, provided the volume to surface ratio of the subsystem is
much larger than the correlation length. This result greatly simplifies the
determination of the ultimate precision of any local estimate of the
temperature, and rigorously determines the regime where interactions can affect
this precision.
