I can't say it's related to GoS but pixel 7 is by far one of the hottest running devices I've ever used in my life. Every now and then it gets so hot I gotta pop it in freezer for a few minutes. It never throttles, it never jitters, but damn does it get so hot I can't hold it.
Comparatively, battery life is... Ok I think? I've seen reports of some users getting 18-24 hours of 'use' but always say "light browsing and texts" uh huh yeah that's why you Installed GoS right? For the sms and browsing privacy...
I personally can get through an 8hr work shift with anywhere from 50%-30% left and usually top up at least once during day. I am a heavy user tho. Music, browsing, emails, games, I like to tinker/test various apps and services and with various things in termux. I also run a VPN pretty frequently (PIA).
Everything on one profile but been wanting to separate. See if it helps. Also, I never used this device with stock ROM. Maybe 1 day then installed GoS. I didn't pay attention to battery life during that time. Very very light use as I had no case yet and I am very clumsy.