What if the size of the Universe doubled?

If you think for a moment about how lengths and speeds in our universe are set (that is, independent of how we choose to measure them, by meters or seconds or whatever), you'll see that these must ultimately come from different ratios of fundamental constants.

I don't know and would be very surprised if there's a way to change these ratios so that all lengthscales or all timescales would change, since so much of what we observe actually comes from very complicated interacting systems acting at different timescales and lengthscales.

So the answer is probably no, there is no sensible physical question behind what you've asked, at least the way you've asked it.

You could ask of course, what would happen if some (dimensionless) physical constant was doubled or halved, and then we could chat some more.

Why do I emphasize dimensionless here? What if you asked "What happens if the speed of light was doubled?" Well, because the speed of light is what sets our time and length scales, we would observe no changes at all! In other words, think about what you can compare the speed of light to that doesn't depend on the speed of light itself. Just saying the numerical quantity changes is meaningless because the units we use to measure it rely (via a possibly long chain of dependencies) on the speed of light itself!

There is no absolute scale to the universe unless you 'turn on' quantum mechanics and general relativity simultaneously. If you do this, then dimensional analysis can tell you that there is a way to use $\hbar$, $c$ and $G$ to construct a unique set of units--the Planck set of units--you can do it yourself pretty easily just by multiplying the three constants together, raised to some power, and trying to get meters, kilograms or whatever out.

So, conceivably, we could tell if were in this other, 'distance squared' universe, by simply doing experiments to measure these three constants, and then calculating the Planck length. If it is different than our Planck length, then it must be some other universe, where we chose a different scale for our distances, or one of the forces is different, or whatever.

Does that answer what you were getting at?

If the universe was flat (which is false), the size would be determined by the age and the speed of light. In other words, it would be a sphere of 13.7 billion light years in diameter. But the speed of light is merely a physical constant which depends on our choice of units.

It wouldn't be twice as large without being twice as old, and obviously this is a big difference :-) In other words, a universe twice as large with the same age, cannot have a big bang, or if you were to measure the age of the universe it would appear to be twice as much as we measure it.