Too large of a AWG can be an issue as can too small. As mentioned, size the wire for the load. I would guess, all that one can do without trouble-shooting, is that there is a thermal protection sensor on the fritz. Maybe a rectifier issue, tube or diode bridge. A transformer issue, maybe, but I would guess that's where the thermal protection would be. A bad solder joint can act up like this, too. Heater circuit on a tube, or bad bias... That's way too many guesses right there.
10 gauge power cord. Too much power for tube amp?
Tube amplifiers tend to be sensitive on incoming voltages. Is there a chance a bigger gauge power cord like 10 gauge may not be a good thing?
My amplifier tends to shut down occasionally upon start up. maybe moving to a higher gauge might be better. Does it matter?