I have a hp pavillion p7-1254
http://h10025.www1.hp.com/ewfrf/wc/document?cc=us&lc=en&docname=c03135882
Often times after having my computer on for not longer than 30 MINUTES the fan will turn on really hard for about 10 seconds then stop. It will do it again a couple minutes later and just stop. It may do this 2-3 times in a row and then not do it for hours. Il feel around inside and its not dusty and nothing in there is warm that i can feel. Why does it do this?I read your topic and I have to say that I'm sorry that you're having this issue. Although I'm not a computer expert just yet, my guess would be that your fan is either failing or is not plugged in all the way.

My questions to you are
Have you overclocked your hardware?

What I mean by this is is your hardware running faster in speed then what it's designed for?

If the answer is yes, then return your hardware speeds to the default
If the answer is no, then continue reading

Is the fan plugged in all the way to the system board. There should be good contact. It's possible that the fan connector is lose.

If the fan looks good, but still does what you claim, then replace the fan and see what happens.

Where is your fan in the computer?If the fan is making a groaning sound which VIBRATES the computer case then its likely bad and should be replaced.

However if it comes on only when the CPU is really busy it is likely functioning as intended with the system set up for variable fan speed controls in which when the system is idle fans can slow or stop spinning and when the system is busy processing data and senses a rise in temp from thermo couple, it will increase a fan speed or TWO to cool the system down.

We had some HP desktop systems at a business which had an air RAM ( squirl cage ) fan mounted above the heatsink of the CPU and when the systems were processing something intensive the CPU fans would wind right up and sound like a jet getting ready for lift off. When the systems were done processing whatever they were doing that was CPU intensive the system would quiet down and idle or operate quietly again.