I've had the same problem with a former laptop. In my case it was an obscure fault: they forgot to put heatsink compound between the CPU chip and the heatsink in manufacture - easy to fix once I'd decided to open it up and take a look. Not something to expect, although I found several other folks who had one from the same batch.
A friend of mine brought me her laptop the other week - it kept shutting down when in heavy use. I could hear the fan wasn't working, so bought a replacement fan off Ebay beforehand. When I opened it up, the internal fan wasn't plugged in to the motherboard. She admitted, then, that it had never been right but she'd been too busy to take it back, and the warranty had expired.
I'd be inclined to download and run a temperature sensing programme to see what's going on. Depending what make/model it is, there may be a specific one for that model, or one of the generic packages such as Everest may do the job. If that shows the CPU going above about 80°C, then something needs to be done.
I have one running all the time on my laptop, displaying CPU and hard drive temperatures (57°C and 36°C at the moment).
The folks above are spot on that a laptop will run faster and hotter when powered off the mains.
Last edited by Keef; 26th Jun 2009 at 23:26.
Reason: To make the formatting work.