Doubling a batch of soup
Kara, you shouldn't need to adjust the cooking time at all. If the recipe says to bring to a simmer and then cook for 45 minutes, it will probably take longer to come to a simmer, but once it is there, you can leave it for 45 minutes.

The best recipes (in my opinion) will give you a time as a guideline, but the real instruction will be some target like "until 160 degrees" or "until peas begin to break down". If your recipe has that, you can look for those cues at roughly the same time the recipe suggests.

edit: I forgot to mention the pot size issue. It shouldn't be a problem. The two issues you could encounter would be:


boiling over
temperature variation


The former you're aware of already; the latter issue is that the soup at the top of the pot could be a bit colder than lower down. The two solutions are:


use a lid (this keeps the heat in)
stir periodically (this keeps both the heat and the peas better distributed)