Life cycle of a gas planet?
The life cycle is just one of gradual cooling and settling. The heavier elements will sink towards the centre; the planet will get slightly more dense - probably reaching about 90% of its current radius. It will essentially become a cold mini-white dwarf made mostly of hydrogen and helium and supported by cold electron degeneracy pressure. The density is probably not high enough for it to crystallise before differentiation of the chemical elements has completed.
The density profile will not change very much from what it is now. If anything the desnsity gradient will become slightly shallower become because a degenerate gas of heavier elements has more electrons per unit mass and so exerts a higher pressure for the same density. Since the thermal conductivity is extremely high then a temperature gradient will gradually be eliminated in the inner regions.
These processes take place on tens of billions of year timescales. Jupiter will be quite similar to the way it is now by the time the Sun becomes a white dwarf.