Is this phone worth repairing?

Hi

I have a Sony Xperia Z5 Premium that died on me a few months ago. I was planning on keeping the phone and going on a sim only when my contract renews in June. Reason being, phones haven't progressed enough past this stage for me to warrant an upgrade.

However, the phone decided to turn itself off and upon turning it back on it stayed on for 10, 20, 30 seconds or so and then turned itself off again. This continuously happened. I managed to do a factory reset in the time it stayed on (plugged in but this didn't always help). The factory reset made no difference.

My theory is one of two possible causes;

1) It's shutting down due to overheating. It gets warm where the processor is, nothing new for this phone and not as hot as it gets when running something processor heavy. possibly just start up warmth.

2) The battery is knackered. It's been replaced once before but I can't remember why I had to do this.

Also, the rubbery compound found on the copper pipes underneath the processor is just that, rubbery. Should this be more like a paste?

Really don't fancy wasting money on parts if it won't fix my phone.

Any help, much appreciated.

Thanks,

Dean