Quote by JesseM
Isn't the whole point just that the density goes to infinity as you approach the big bang, and the distance between any two points in the universe which are today some finite distance apart goes to zero? This isn't the same as saying that "the universe starts at a particular time in a relatively small geometric region", unless you're just talking about the observable universe. Would you agree that in the standard FriedmannRobertsonWalker cosmological model, if the universe is flat or open then its volume is infinite at all finite times after the big bang?

I think the distinction you are coming to is a semantic one. In other words, how the phrase "volume of the universe" defined determines the answer, and I believe that at least two different definitions of that phrase are being used in this case.
I would intuitively define "volume of the universe" operationally as something on the order of: "(1) select two points at which matter or energy arising from the Big Bang are present, which are as distant or more distant from each other than any other two points in the universe; (2) call the magnitude of the distance between them d; and (3) the volume of the universe in the spacelike dimensions is then defined to equal pi*d/6".
In a conventional Big Bang scenario with radiation emitting in all directions from day one and outpacing everything else, one would expect that d would be approximately equal to 2*c*t, or in speed of light units simply 2t, so long as the universe is not contracting. Hence, in a Big Bang scenario, the 3D volume of the universe, if this definition is adopted, is a function of the time elapsed since the Big Bang (defined as t=0 and hence the volume of the universe overall through point t in four dimensions would be the integral from zero to t of f(t) with respect to t. Hence, this definition would produce a finite 3D volume of the universe at any given time t, and a 4D volume of the universe that is infinite or finite depending on the form of f(t) (which depends on the values you put into the FriedmanRobertsonWalker equation in standard GR). (Of course, one would have to be quite clever in defining "t" in the equations above in a way that makes sense).
This isn't quite the same as the "observable universe" (and certainly less elegant) although it is pretty close.
It sounds like the definition of "volume of the universe" you are using is something on the order of V=pi*d/6 for the value of d (defined as above) that is the maxima of the function d(t)= for t between 0 and infinity. This would be infinite given the proper inputs into FRW.
The implicit issue that hinges between the two definitions of "what is the universe whose volume we are measuring" is whether empty space should be included when you are defining what the universe is. In a nonaether theory, it would seem to make sense not to include that empty space. In an aether theory, it is vital to do so. General relativity, is basically a nonaether theory that gets a close to an aether theory as it is possible to do, because its geometrical elements are very aetherlike.