We study the thermal stability of the periodic (P) and quasiperiodic (Q) precessional modes of the nanoparticle magnetic moment induced by a rotating magnetic field. An analytical method for determining the lifetime of the P mode, in the case of high anisotropy barrier and small amplitudes of the rotating field, is developed within the Fokker-Planck formalism. In general case, the thermal stability of both P and Q modes is investigated by numerical simulation of the stochastic Landau-Lifshitz equation. We show analytically and numerically that the lifetime is a nonmonotonic function of the rotating field frequency, which, depending on the direction of field rotation, has either a pronounced maximum or a deep minimum near the Larmor frequency.