The Higgs field was not spontaneously broken when the universe was young and hot (like I am not...:)). At the very high temperature that existed back then, the Higgs' energy of the Higgs field was non-zero. Only after the alleged symmetry breaking the field attained a zero value.
Where did the energy of the Higgs field (Higgs particles with energy/mass) go? There must have been a lot of Higgs particles. If we look at the Higgs Mexican hat potential we see that around the zero field there are a lot of finite energy potentials associated with different field variables. For a zero field there is even a maximum energy.
How did these real particles (contrary to the virtual ones alledgely existing today which don't contribute to the energy in spacetime) contribute to the energy of spacetime? Did they constitute the inflation field which made matter move away from each other? Did they give rise to a negatively curved spacetime on which the initial particles rushed away from each other?