The way I understand spontaneous symmetry breaking in thermodynamic systems is that the symmetry is actually explicitly broken by an infinitesimally small field. The system chooses one of the non-symmetric states effectively at random because it is infinitely sensitive to the smallest of perturbations. For example, in the case of the Ising model, the magnetisation depends discontinuously on the exernal field,
$$m(h\rightarrow 0^+) = m_0 \neq m(h\rightarrow 0^-) = - m_0 \, .$$
There are two possible values of the magnetisation in the symmetric case, $h= 0$ and the physically realised one depends on the sign of an infinitesimally small (and not measurable) external field. It appears random.
What about cosmology? Everybody talks about electroweak symmetry breaking during the early stages of the big bang, for example. What is meant by that? The big bang (and the whole universe) is a non-equilibrium system. I don't know how to relate it to thermodynamics. Moreover, in cosmology, there can be no external field to infinitesimally break the symmetry. How can the symmetry really be broken at random? Is the universe not deterministic?