I'm no expert, but you might try to isolate the source of the problem with a couple simple experiments.
iCloud: It has been my experience, that when something is corrupted in the cloud (e.g. you Contacts) preventing a successful sync, your computer will work very hard to repeatedly attempt a sync.
Thunderbird/DisplayPort: There are been a number of reports of Thunderbird port causing the CPU to peg at 100% when a device is attached.
Turn off internet (unplug ethernet and turn of WiFi). Then leave it running as you describe, and see if that changes anything.
Unplug everything from the Thunderbird (if there is anything). See if that changes anything.
These experiments don't solve any problem, but they might put you on the right track. Ultimately, you may need to go to the Apple Store and have them take a look (make an appointment with Genius Bar). The more you can tell them, the easier it will be for them to figure out what's going on.
I hope this is helpful.
Thanks! I will definitely try that. I actually shut the screen saver off and left it for the past few hours with out any problems so far. I wonder if it was that? I did notice that it always shut off when it was on the same saver (it was set to "random"). I'll give it a few days and see if it happens again. Thanks again!