It is the energy scale of interactions that controls whether the three forces are unified and when symmetry has been broken. The temperature of the universe as it expands gives also the average kinetic energy of the elementary particles that are contained in the universe at that time. That is the energy with which the particles will scatter and interact and produce new particles .
Here is the Big Bang time line:
The unification scale is controlled by the coupling constants: energy behavior
Plot shown is with assumed supersymmetry, without that assumption the meeting point is not clean.
Here is a simplified diagram that shows how symmetry can be broken given the shape of the potential.
Spontaneous symmetry breaking simplified: - At high energy levels (left) the ball settles in the center, and the result is symmetrical. At lower energy levels (right), the overall "rules" remain symmetrical, but the "Mexican hat" potential comes into effect: "local" symmetry is inevitably broken since eventually the ball must roll one way (at random) and not another.
The effective potentials for the interactions depend on the coupling constants.
The strong force has a different energy behavior and separates first, as the temperature falls and the available energy of interactions decreases, which gives a window for a quark gluon plasma phase. Then with the expansion the interaction energy available falls to the point where electroweak symmetry breaking happens.
So the timeline is fixed by measurements of the coupling constants in the lab and the use of theoretical calculations that describe their energy dependence.