In the chamber, the hot gases are >3500K, and touching the inner walls with a melting point at ~1400K, yet the walls don't melt, because they are cooled with more power than they are heated.
I'd argue that they're being cooled with equal power, otherwise they could freeze infinitely which is physically impossible.
[1]: https://arstechnica.com/science/2020/05/nasa-will-pay-a-stag...
[2]: https://www.google.com/search?q=merlin+engine+dry+mass
[3]: https://www.wolframalpha.com/input/?i=%28630+kg+of+gold+in+U...
https://www.wolframalpha.com/input/?i=%28%28%28630+kg+*+%28d...
Current spot price of gold is $1800/oz. Merlin dry weight is 1380 pounds. 1380 pounds of gold is right at 40 million dollars.
Per https://en.wikipedia.org/wiki/RS-25#Space_Shuttle_program :
"A total of 46 reusable RS-25 engines, each costing around US$40 million, were flown during the Space Shuttle program"
Beautiful.