Jump stat car but won't stay running - 1994 Buick LeSabre

Check the battery, either it is nearing the end of it's lifespan or something is draining the battery. Try another battery and see if the problem persists, if so, you will need to find the cause of the battery drain, could be a trunk light staying on or some other unseen fault

You can use code scanner to check the engine system first.

you may have a bad ground, or the ignition switch needs to be replaced.

Car starts fine with a jump? You sure your battery is good? If the battery is good then you must have a connection bad right at the battery. If you jump it and it starts then the starter is ok. If you jump it and it stays running with lights on, heater fan on, etc. then the alternator is probably ok. The alternator won't keep the car from starting, but it might not let it start the next time if it is not recharging the battery. Please rate my answer so I am rewarded points for it.

If is fuel injected try checking the fuel regulator and if is has a carburetor possible bad choke or needle seat / worse case jump timing

If it dies after you remove the jumper cables, then the alternator is not charging.

If you can jump start it and it stays running....I am assuming that you connect the Jumper Cable Ground to something other than your battery. If so, Your negative battery cable is suspect. Check both ends for ANY corrosion or loose connections.

Clean all three battery connectors and replace bolts

It sounds like air in the system if it was the thermostat it would not fluctuate.

could be faulty battery or alternator not charging