  We show that the third Goldstone mode, which emerges in binary condensates at
phase-separation, persists to higher inter-species interaction for density
profiles where one component is surrounded on both sides by the other
component. This is not the case with symmetry-broken density profiles where one
species is to entirely to the left and the other is entirely to the right. We,
then, use Hartree-Fock-Bogoliubov theory with Popov approximation to examine
the mode evolution at $T\neq0$ and demonstrate the existence of mode
bifurcation near the critical temperature. The Kohn mode, however, exhibits
deviation from the natural frequency at finite temperatures after the phase
separation. This is due to the exclusion of the non-condensate atoms in the
dynamics.
