Your specific question is about why uniform gas is a low entropy state for the universe. The reason is that you make entropy by allowing the gas to self-gravitate and compress, releasing heat to the environment in the process. The end result is a black hole where the gas is compressed maximally, and these are the maximum entropy gravitational states.
But the uniform gas comes from a nearly uniform inflaton field shaking over all space after inflation. This inflaton produces uniform denisty of matter, which then becomes uniform baryons and hydrogen. Ultimately, it is the uniformity of the energy density in the inflaton field which is responsible for the low entropy of the initial conditions, and this is linked to the dynamics of inflation.
The dynamics of inflation produce low entropy initial conditions without fine tuning. This seems like a paradox, because low entropy is fine tuning by definition, don't you need to choose a special state to have low entropy? The answer in inflation is that the state is only special in that there is a large positive cosmological constant, but it is otherwise generic, in that it is a maximum entropy state given the large cosmological constant.
The theory of inflation explains the specialness of the initial conditions completely. This was proposed by Davies in 1983, but it is rejected by cosmologists. The rest of this answer discusses arguments that support Davies' position.
deSitter space
If you consider a deSitter space with some mass density added, and you look in a causal patch (meaning what one observer can see), the mass density gives an additional curvature without (significant) pressure and turns deSitter into more like a sphere. There is a continuous deformation of deSitter space into the Einstein static universe, which is obtained by making the density of matter as large as possible.
Any matter you add reduces the horizon area of the cosmological horizon, and this is true for black holes as well. If you consider the ds-Schwartschild exact solution, for example, you can have an isolated black hole in deSitter space:
$$ \rm ds^2 = - f(r) \rm dt^2 + {\rm dr^2\over f(r) } + r^2 \rm d\Omega^2 $$
$$ f(r) = 1 - {2m\over r} - {\Lambda r^2\over 3} $$
but there are two horizons, and the causal patch is the region between the black hole and the cosmological horizon. It is easy to check that the total horizon area, cosmological plus black-hole is maximum for m=0. It is also easy to check that there is a certain value of m where the black hole radius and the cosmological radius degenerate. At this degeneration, the distance between the black hole and cosmological horizon stays constant, they do not collide except in the bad r coordinate, and the space turns into AdS_2 x S_2.
Nariai dynamics
Imagine starting near a Nariai solution with additional matter between the two horizons. These are both still black holes, neither is a cosmological horizon, as you can see by adding more matter with a uniform density, until you approach the limit of the Einstein static universe with two antipodal black holes.
This is a physical configuration of the static cosmology. So you can start with an Einstein static universe, and evolve it forward in time, you will produce black holes, and they will merge and grow.
If you take all the matter in the static universe and push it into one of the black holes, this black hole area will increase past the Nariai limit and it will become the cosmological horizon. At this point, the singularity runs away to infinity. If you push the matter into another black hole, the other black hole will be the cosmological horizon. It's up to you.
So if you start with the Einstein static universe, the black holes compete for mattter, until eventually the biggest black hole will surround all the others, and become the cosmological horizon.
The lessons are the following:
- Cosmological horizons are the same stuff as black hole horizons. Their other side is described by black hole complementarity, just as for black holes. It is wrong to think of the universe in a global picture.
- deSitter space is the maximum entropy configuration of a positive cosmological constant universe, everything else eventually thermalizes into deSitter space.
- The global picture of black holes is not particularly physical, because the singularity of the Nariai solution runs away to infinity in the Nariai limit. There are cases where black hole interior structure degenerates.
Inflation Produces Low Entropy Initial Conditions
The second point answers your question, because the early universe is in a deSitter phase. So given a large positive value of the cosmological constant in the early universe, the maximum entropy state is a deSitter space with a cosmological horizon of small area, and this is necessarily a low entropy initial condition for later times, during which the cosmological horizon grows.
There is no further explanation required for the low-entropy initial conditions. This is the same explanation as for all the other miracles of inflation, the killing of fluctuations, the flatness condition, the monopole problem. The whole point of inflation is to produce a theory of low entropy initial conditions, including gravity, and it does so naturally, because deSitter space is the only low entropy maximal entropy gravitational state. This answer was first given by Davies, and it is just plain correct.
This plain-as-the-nose-on-your-face idea is not accepted despite the nearly thirty years since Davies' paper. I should add that Tom Banks and Leonard Susskind both now say similar things, although I don't want to put words in their mouths.
Best Answer
They are one and the same, and the opposing model is the Cold Big Bang.
Also see the Big Bang page of wiki, it also has clues to why it may be called hot.
Cold Big Bang