Dunno about same tone cold vs hot days.
But I know there's some sort of stick-on you can get to adhere to your lip plate. It's not permanent, and I guess does wear out. But it's supposed to stop slipping. Sorry I can't be more definite, I don't think I ever had slipping issues so didn't pay a lot of attention to the thing. More of a "oh, ok, interesting, move on" kind of passing notice. I would guess it's on some flute site--maybe flute specialists? Or Carol Naubaum (however you spell that?) Or flute world? Those are the only sites I know I've been to. Again, sorry I don't remember better, it's been several years since I looked and bought my bass flute's gel pad.
1975 Gemeinhardt M2 in chrome nickel
1982 Armstrong 80
2006 Yamaha 584
1978 Artley piece of crap 15 P
1982 Gemeinhardt 4S
1980s (?) Armstrong all wood (no model)
2006 Jupiter di Medici G0199
Plus many many flute-cousins....