Read from this issue as well in the German News, so I guess you are not alone.
You could try those troubleshooting steps to try to fix the issue, but if this is still the case I would recommend to either reach out Stadia or more likely Chromecast Support to report this problem and get more steps or a fix.
Yes, I made a thread about it earlier there seems to be a overheating issue going on where people play for a few minutes and it resets on them, I saw on reddit as well that it was a common problem with chromecast but i feel this is much worse especially with heavy gaming.
My Chromecast Ultra just overheated after playing Destiny 2 for only a couple of hours.
I haven’t had the same overheating issue again but it is still getting hot to the touch after playing for a couple hours. I was seeing that this is a normal issue for chromecast to overheat like that just with regular streaming. I was thinking about getting a small PC fan and connecting it to try to keep it somewhat cooler.
I found this on amazon, it’s only $10. I might try it if it overheating again.
https://www.amazon.com/dp/B076DN1WX8/ref=cm_sw_r_cp_api_i_MlW2DbFFAW9FG