  The Lipkin-Meshkov-Glick model is used to examine the validity of some
approximate methods in a many-body theory at finite temperatures. Namely, the
thermal random phase approximation (TRPA) and the thermal renormalized random
phase approximation (TRRPA) are studied. An average energy of the system, an
average quasispin projection and a particle number variance are calculated
within these approximation and exactly with the grand canonical ensemble
partition function. On the whole the results of TRRPA are found to be in better
agreement with the exact ones. The validity of the both approximation becomes
better with increasing temperature as well as particle number.
