I have a 2008 Honda Civic and I started hearing a loud exhaust sound when I back up and when it is cold. What could it be?

this is going to require an inspection to determine which parts need replacement...engine mounts are a good guess....get a woodblock and floorjack to see how much travel there is in those mounts~ hopefully will NOT be a cracked exhaust manifold~

I was thinking along the lines of an exhaust manifold as well. Might explain why the vehicle is louder while cold and when the engine warms up the expanding metal will close the gap making it sound...well, not as loud. A very common problem is that bolts tend to loosen up over time and the manifold will just loosen from the engine block. When the vehicle has sat for a while and cooled, see if you can find a wrench of the correct size for the bolts and try to tighten them down starting from the center of the manifold and work your way outwards toward the opposite ends. By doing this you might be able to pinch the manifold down a little tighter and eliminate the sound all together. If this doesn't work, I would suspect a crack and that means replacing the manifold. Good luck.