  In this paper we develop a quantum field approach to reveal the thermodynamic
properties of the trapped BEC with the equal Rashba and Dresselhaus spin-orbit
couplings. In the experimentally-feasible regime, the phase transition from the
separate phase to the single minimum phase can be well driven by the tunable
temperature. Moreover, the critical temperature, which is independent of the
trapped potential, can be derived exactly. At the critical point, the specific
heat has a large jump and can be thus regarded as a promising candidate to
detect this temperature-driven phase transition. In addition, we obtain the
analytical expressions for the specific heat and the entropy in the different
phases. In the single minimum phase, the specific heat as well as the entropy
are governed only by the Rabi frequency. However, in the separate phase with
lower temperature, we find that they are determined only by the strength of
spin-orbit coupling. Finally, the effect of the effective atom interaction is
also addressed. In the separate phase, this effective atom interaction affects
dramatically on the critical temperature and the corresponding thermodynamic
properties.
