I have this same problem, power comes on for a few seconds then computer shuts down. Brought it in for repair and the Hard Drive is fine, as suggested I pulled the HD and attached a cable and saved all my data ^
I sent the computer out and will post back when they respond. I think it may be the power supply (internal) and the cooling system.
I see I am not the only one with this problem

My money's on a bad motherboard on Vulcan's.

But his is hanging, not shutting down.

Coming on for a second or two and then shutting down is usually a power supply issue.

(Oh, wow, my first post on here, LOL)

Anyways, based on my experience with dealing with unwell motheboards, I'm suspecting one of two things:

a) Motherboard problem, specifically problem with capacitors or heat sensor on motherboard (some HPs, but particularly Sony Vaios, are bad about caps on the motherboard going bad)

b) *POSSIBLY* an issue with the heatsink not providing enough cooling (either cooling fan not operating or heat compound needs to be replaced)--this is by far a less expensive fix ($5 verses $100 or more depending on age of PC).

(With the laptop starting up then shutting down--as opposed to staying on but not showing any video, or not powering up/erratically powering up--this does point to it being more of a problem with the PC thinking it's overheating)

Depending on your budget, there are a few places that do specialise in this sort of repair, including http://www.ikenfixit.com (usually he will do repairs for $50-300 depending on what exactly needs to be done, with power jack and cap replacement being cheaper).

Another possibility re HP laptops in particular--many HP laptops tend to have the power jacks become loose over time (due to the power cords used putting excessive stress on the jacks), and if this is not corrected in due course this can take out the entire power stage of the motherboard (I've recently had to replace a motherboard on an HP ZE4500 because of this issue).

One good thing on HP motherboards--generally you can find verified working motherboards for around $150 or so for more recent kit and cheaper for older machines, if you either feel adventurous in installing them or know someone who feels comfortable with working with laptops. (Vaio mobos are horrifically expensive--around $500 or so for Pentium III machines; Toshibas are among some of the more economical to restore :))

(Yes, for the record, I've repaired PCs and laptops for work and for fun :3)