As I understand, from watching the Discovery Channel, the total amount of energy in the universe is zero. As such, people like Hawking explain that the universe can be created out of nothing because... well it actually is nothing. That's fine, but if that is true today I guess it also was true shortly after the Big-Bang. This period is usually described as being extremely hot and dense and at the very moment of the Big-Bang the energy density is described as being infinite. How can this be true as the total amount of energy at that time was actually zero? Somehow the description hot and dense doesn't match very well with zero energy. What am I missing here?
In the standard homogeneous cosmological models the total energy in an expanding volume is zero. This is true for positive, negative or zero curvature and it must take into account the gravitational energy (which is negative), dark energy, matter and heat. Since the gravitational energy is negative the heat can be positive and increasing as you go back towards the big bang.
However, "can be" is not the same as "must be". The other possibility is that the energy in the vacuum during the period of inflation counterbalances the negative gravitational energy. The universe before or during inflation could then have been cold and heat would have been released at the end of inflation when the vacuum collapses. Heat would also be generated by matter annihilation. Again this is only one of many possibilities. We can't say if the universe started hot or cold. we only know that it heated at some point very early on. Either possibility is consistent with zero total conserved energy.
In Zero-Energy Model, negative energy associated with Gravity counterbalances positive energy associated with matter, photons, etc. So, No, Big Bang wasn't cold. You are just looking at partial picture (you just ignored Gravity).
This is what Zero-Energy Model says:
With traditional Big Bang model (which doesn't contain Inflation), the universe started out with zero energy (but hot) and continue to exist with zero energy till today.
With Inflatory Big Bang model, energy from gravitational curvature transferred to matter (and other positive matter contributors) in order to make the total energy of the universe zero, if it wasn't already zero. All several phases of traditional Big Bang are shifted to post Inflation (making end of Inflation as starting point of Big Bang evolution) containing the total energy of phases zero as that of traditional Big Bang model. Inflation and post-Inflation phases were hot. For before Inflation, the model provides nothing, but as for temperature, most physicists believe that the universe was hot. There are cold Big Bang models in existence which exploit this lack of information. But, even with cold Big Bang models, total energy of post-Inflation universe is zero (till today).
Note: Zero-Energy Model may not be correct as it's more or less a kind of hand-waving model. In non zero energy case, it shouldn't be problem seeing hot Big Bang, I suppose. In reality, energy of universe isn't absolute or constant and should not be definable, smashing Zero-Energy Model.