If you want it not to get hot, the CPU is probably of secondary concern. Get an efficient POWER SUPPLY first, then worry about the CPU.
Power supply efficiencies vary between the mid to low 60% range up well into the 80% range. Considering this is working on ALL the power required for your PC, not just your CPU, signifciant improvements here, can lead to a larger overall reduction in heat load than anywhere else. Most cheapie power supplies are definitely in the LOW efficiency range. To get a supply in the 80% range you're typically paying more for the same Wattage of power supply, and most people will not do this. Efficiency just doesn't seem to sell nearly as well as price per Watt.
http://www.silentpcreview.com
has AWESOME power supply reviews that focus on noise and efficiency.
Now you probably don't need exceptional speed for what you're doing, so I'd suggest you get an AthlonXP Mobile and a decent nForce2 motherboard. Run the mobile close to stock speed with 200 FSB (they are all stock 133 FSB, but change the FSB to 200 and the multiplier to get it close to stock speed, as this will yield significantly better RAM performance) and you'll have a pretty snappy system with the processor using less than 45W (the Prescott core P4s can run closer to 80-100W). You won't be breaking any speed records, but it would be reasonably inexpensive, and definitely efficient. The A64 mobiles are a little less supported in desktop boards, while the AXP mobiles are pretty much universally supported in desktop boards.
The TDP (Thermal Design Power) of Intel and AMD processors are published on their site, but interpreting them is more difficult than it used to be, as they typically only publish the maximum power of the fastest processor, and give you little idea of what lesser processors will use (obviously less power than the fastest).
Celeron D processors are rated at 73 W TDP, Prescott P4s are going to be higher.
A64 processors are rated at 89, but the 'intermediate state at 1800 MHz / 1.4v (same as an A64 2800+) is rated at 67W
Athlon XP Mobile processors are rated at either 35W or 45W, depending on speed.
So I'd say you're looking at deciding between an AXP-Mobile, a Celeron D or an A64 2800+. I'd choose the 2800+ and a nice efficient power supply.
For reference, the 0.13 micron 1000 MHz P3 was under 30W, but the older PIII 1GHz was closer to 45W.