  We discuss a class of mechanical models of thermometers and their minimal
requirements to determine the temperature for systems out of the common scope
of thermometry. In particular we consider: 1) anharmonic chains with long time
of thermalization, such as the Fermi-Pasta-Ulam (FPU) model; 2) systems with
long-range interactions where the equivalence of ensembles does not always
hold; 3) systems featuring absolute negative temperatures. We show that for all
the three classes of systems a mechanical thermometer model can be designed: a
temporal average of a suitable mechanical observable of the thermometer is
sufficient to get an estimate of the system's temperature. Several interesting
lessons are learnt from our numerical study: 1) the long thermalization times
in FPU-like systems do not affect the thermometer, which is not coupled to
normal modes but to a group of microscopic degrees of freedom; 2) a thermometer
coupled to a long-range system measures its microcanonical temperature, even at
values of the total energy where its canonical temperature would be very
different; 3) a thermometer to read absolute negative temperatures must have a
bounded total energy (as the system), otherwise it heavily perturbs the system
changing the sign of its temperature. Our study shows that in order to work in
a correct way also in "non standard" cases, the proper model of thermometer
must have a special functional form, e.g. the kinetic part cannot be quadratic.
