Replacing the thermostat will probably not affect performance on a stone cold start up. After the engine reaches correct operating temperature, (~195F) is where you'll notice a difference, guaranteed. When the engine operates at less than 195F, the PCM adjusts fuel mixture richer than it does at 195F. So, if this temperature issue has been going for a while, the oxygen sensors are probably carboned up a bit, which slows their reaction time down. 20 or 30 miles on the freeway should clean them up. After that, you might want to stop at a shop and have them retrieve the DTC's. Having them handy will help in diagnosing any other problems.

Yeah, that's kind of what I was thinking. The engine will be cold at start up, regardless of whether or not the t-stat is working or not. I'm just hesitant to drive it to a shop because it isn't registered or inspected, but I'm sure having the codes would help diagnosis the problem.