|MadSci Network: Astronomy|
Yes, you could say the "volume" of the observable universe is increasing... but for two very different reasons!
One reason has nothing to do with the expansion of the universe, but simply its age. The best guess is that the universe is about 13 billion years old, which means that we can only see 13 billion lightyears in any given direction. Looking further than that is impossible, because you'd be looking back at a time before the Big Bang! So the universe is getting "bigger" over time, simply because we can see more of it. Every second, the observable universe becomes one light-second (300,000 km) bigger in each dimension.
Next comes the fact that the universe is expanding. The best measurements of the Hubble Constant show that new space is being created at a rate of 70km per megaparsec every second. This means that for every cubic megaparsec of space (1 megaparsec = 3.26 million light years), the volume increases at a rate of 70x70x70 = 343,000 cubic kilometers per second.
However, there's nothing for the universe to do work on; despite the expansion, the total energy in the universe stays the same. As galaxies get further apart, the gravitational potential energy becomes less negative (it increases), but the galaxy's kinetic energy becomes less (the galaxies slow down a bit) keeping the total energy the same. If there's really some negative vacuum energy that is accelerating the universe, then this picture changes a bit, but there's still nothing for the universe to do work against.
Here's a useful cosmology page.
Try the links in the MadSci Library for more information on Astronomy.