I had a P4 with HT @ 2.8 Ghz in one of these, if it counts for anything. It just died last year. Heat definitely wouldn't be a problem. That cooler idles(idled) at like 35C. The stock PSU, however, wouldn't be able to handle a GPU and a P4 at that speed.
I hope this helps you in your decision.
Pentium AE @ 4.3 Ghz, Asrock Z97 Pro4, 16 GB of DDR3-1600L, 250 GB 850 Evo, 750 GB HGST 2.5'', Gigabyte 750Ti OC Edition, Corsair RM750, Corsair 550D, Windows 10 Pro