If the universe is flat, does that imply that the Big Bang produced an infinite amount of energy? Too much density and the universe is closed, analogous to a sphere in four dimensions: you travel in a straight line and you end up where you started.  Too little and you have a saddle: not sure about the destination if you travel in a straight line.  Just the right amount and the topology is flat.  The flat topology is infinite: you travel in a straight line forever.
If the topology is flat (and at this point all evidence indicates that it is to within 0.4%), then multiplying the critical density by an infinite amount of cubic meters gives you an infinite energy/stress.$$\rho_{CRIT}\space kg\space m^{-3}\times \infty\space m^3=\infty\space kg$$
Is that a reasonable interpretation?
 A: What you calculated above ($\infty$, for a flat space universe) is just the proper mass-energy of matter.  It doesn't take into account the - negative - energy of the gravitational "field" itself, which cannot be localised in General Relativity.  The "total energy" of the universe could be 0, but there's no way we could give a physical sense to it, since the whole universe energy cannot be measured from "inside".
A: You have to be careful with the interpretation of energy when you don't know what sort of gravitational potential well might be involved. For example, you might find some amount $m c^2$ of rest-energy of matter when calculated in an inertial frame near but outside the horizon of a black hole, but in order to use this energy at some other location, you would first have to pull the matter up against gravity, expending energy $E$ in order to do so. After spending that $E$ you acquire just $m c^2$ at your location, so overall you have gained $(m c^2-E)$ and this will be small compared to $m c^2$ if the matter started out near a horizon. This is the sense in which gravitational binding energy is negative. When applied to the whole universe, this consideration makes a calculation of the type you are proposing questionable, because it is hard to say what physical meaning it has.
A better analogy is, perhaps, with the concept of escape velocity. A flat universe is one where the motion of matter everywhere is just enough to keep escaping from its own mutual gravity.
Finally, the topology of a mathematical space is not in one-to-one correspondence with the curvature, and in particular, if a space is flat it does not necessarily follow that it is infinite. There are a number of different topologies that are mathematically possible for a flat space, and some of them are bounded (i.e. not infinite). So this may apply to the physical universe too. We don't know.
A: If it was completely flat, you wouldn't talk about volume. The universe isn't completely flat, it does have a volume (like a pancake, it's called 'flat', but it definitely has a volume). For a truly flat object, your formula would need a surface density and surface area, and you would get a finite answer.
