  We derive explicitly the thermal state of the two coupled harmonic oscillator
system when the spring and coupling constants are arbitrarily time-dependent.
In particular, we focus on the case of sudden change of frequencies. In this
case we compute purity function, R\'{e}nyi and von Neumann entropies, and
mutual information analytically and examine their temperature-dependence. We
also discuss on the thermal entanglement phase transition by making use of the
negativity-like quantity. Our calculation shows that the critical temperature
$T_c$ increases with increasing the difference between the initial and final
frequencies. In this way we can protect the entanglement against the external
temperature by introducing large difference of initial and final frequencies.
