Is it in a lamp, or a recessed lighting fixture, or what? In any case, you might consider replacing the socket. Perhaps the contacts aren't very good and you are getting heat buildup, which is shortening the life of the bulbs.

Yes, try replacing the socket as Kudzu suggested. In a prior home, I had a range hood that ate light bulbs--a new bulb would last only a couple of months. Decided to replace the socket--which was very easy to do. Problem solved.

Not all LED bulbs are rated for use in enclosures, heat will build up and shorten the bulb life. The suitability will be marked on the packaging.

I got it apart this far. I'm not sure how to proceed. Of course, I turn the lights off when I go in there

Without a close up picture this is probably what you will have to do: Need to disconnect the wires to that socket up there in the hub. After you get the wires loose, I would wire a slender flexible (guide) wire to the loose electrical wires before pulling the socket out of the fixture. You will leave that guide wire in place and use it to pull the wires on the new socket back through the curved arm on the fixture.

You can buy a special lubricant that is used when pulling wires thru conduit. Would help to have some of that. Or use any dry spray lubricant that you might have on hand. Inside the tube and on the socket wires before trying to pull them.

You can disregard the comment about enclosed bulbs, if you go with ci_lantro's suggestion and make the connections correctly you should be ok. I don't think lube is necessary for a short run like that when you use a pull wire.

Before replacing the socket, I’d try pulling the little tab at the bottom of the socket up. It may have gotten bent too far down over the years and could be the cause of the loose connection and overheating.
It might be a quick and easy fix. Turn the power off when you are doing it (at the breaker box, not just the light switch)

Check first with a tester to make SURE there is no voltage where you are working!

I second bending up the tab in the center of the socket.

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