At what energy do up and down quarks unbind? At what energy do up and down quarks unbind? Google does not seem to have a clear answer. Could someone please represent this mathematically?
 A: Quarks do not unbind in the low energy regime. They are color charged, and thus confined1, the strong force between them does not fall off with distance (as is the case for the electromagnetic force with $\propto \frac{1}{r^2}$), but increases linearly with distance. Thus, you need ever more energy to pull two quarks apart, which is stored in "flux tubes" (essentially the strong field lines) spanning between the quarks. When the energy stored in the "tension" of these flux tubes2 exceeds that needed to create a quark-antiquark pair, they will sooner or later break up, and you'll have two pairs of quarks connected by shorter flux tubes then. You cannot ever have a free quark. 

1No rigorous proof of confinement exists, though you can see confinement arising in some lattice models and through Polyakov loops on tori. QCD can be seen to have confining and deconfining phases, and the confining phases occur at the energy/density/temperature scales that are normal for our observations.
2Some models treat the flux tubes as strings with tension.
A: The other side of the coin then ACuriousmind 's answer is the quark gluon plasma. The quark gluon plasma 

is a phase of quantum chromodynamics (QCD) which is hypothesized to exist at extremely high temperature, density, or both temperature and density. This phase is thought to consist of asymptotically free quarks and gluons, which are several of the basic building blocks of matter.[citation needed]. It is believed that up to a few milliseconds after the Big Bang the Universe was in a quark–gluon plasma state.

Within the plasma the quarks are asymptotically free

Asymptotic freedom is a feature of quantum chromodynamics (QCD), the quantum field theory of the nuclear interaction between quarks and gluons, the fundamental constituents of nuclear matter. Quarks interact weakly at high energies, allowing perturbative calculations by DGLAP(name of equations) of cross sections in deep inelastic processes of particle physics; and strongly at low energies, preventing the unbinding of baryons (like protons or neutrons with three quarks) or mesons (like pions with two quarks), the composite particles of nuclear matter.

They are measuring quark gluon plasma at the LHC, at least fitting the data with models : Search for "quark gluon plasma lhc", as they can go to small distances/high energies.
The crux is "small distance high energy" . Thus quarks will always be confined at the energies we can achieve in the laboratory, the bag becomes bigger but we cannot see an individual quark. At the Big Bang energies, where there is a soup of free quarks and gluons, it is just that the bag becomes the whole universe at the time, and we will not be there to do experiments :).
In the BB chronology the quark gluon plasma is hypothesized between 10^-32 seconds and 1 second after the BB, in this plot the energies are between  10^14 GeV and 1 GeV. (These are just indicative numbers, and depend on the particular model.)
