Notebookcheck.net, despite the name, also reviews phones. They use an IR camera to check the temperature of the phone in different areas, so you can see how specific phones handle under load.
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Check for the chipset used on the phone. Usually phones that only uses ARM Cortex A55 cores never got hot and have longer battery life. They are usually pretty slow though. On the other hand, phones that have one or more ARM Cortex A7x cores (A75, A77, etc) in addition to A55 cores typically run hotter but run apps faster, especially when the A7X cores are configured to run at high clock rate (>2.5ghz).
For comparison, your old pixel 4a has 2 cortex A76 (2.2ghz) and 6 Cortex A55 cores, so I imagine it doesn't run very hot. Generally, the lower the clock rate and the fewer A7X cores a phone use, the cooler it is.
There's also the manufacturer to consider, the SD8gen1 and SD8gen1+ are basically the same chip, but made in samsung and tsmc fabs respectively. The tsmc made chip runs A LOT more efficiently.
In addition to what was said in the comments and at the risk of stating the obvious, look also into minimizing the number of apps that keep working in the background and aren't really idle (e.g. background data collection) and replace as many apps as you can with opensource equivalents that aren't data hungry and don't hide any nasty background telemetry. If the CPU is not idle then it will get hot.
Also, the battery saver should usually help you get lower temps.
Most of the time it depends on the processor. Some processors are notorious for using a lot of energy and therefore get hot. It's hard to know which ones exactly don't get hot but generally I would binge reviews and avoid bad processors.
Read GSM arena reviews, they test for temperature. Look for 600 series snapdragons or similar.
Aside from that, any high end phone chases power over efficieny. If you can deal with something bigger, more phone will mean an overrall cooler surface.
You can also look at specific SOCs to see how users are liking them. The SD855 was the last chip that I know was very balanced between power and efficiency. The SD8gen1 is a hot bastard, but now the SD8gen2 seems to have ended up another magic chip that runs cool despite its power.
I’d say the 865 was the last “good” Qualcomm chip before the 8 gen 1 broke everything. It can be found in the Galaxy S/Note 20 generation.
Pretty much any soc made by samsung is hot and slow. The return to TSMC with the 8+ gen 1 we saw the thermal improvements.
8gen1 was absolute trash. Performance was phenomenal but only until throttling murders it. It was so bad, active cooling became a thing in phones. Ridiculous.
It killed sonys Xperia mkIVs. Only the 10, which was on an SD695 was any good. 695 seems like another good one, actually. A sibling and my mom got phones with that SoC, and the phones work fast, while sipping battery.
8gen2 seems like a return to form. TSMC again, surprise surprise.
I still daily a SD865 (LG V60) to this day and while it's losing steam in newer mobile gaming titles, it's still flawless for daily usage. Can't think of any complaint really.
Buy something older or lower-end. Most high-end flagships these days run hot.
Do you have any particular recommendations?
Unfortunately generally the "slower" chip sets also run cooler so many flagships will have heating issues to contend with depending on ambient temperature.
I hate my P6 Pro for that reason, and will be doing the same as OP and avoiding any poorly heat throttling phones in the future.
I'm tired of having my videos shut off mid shoot because of heating issues, or issues while using GPS on the motorcycle.
If you wanna run cool, you gotta run on heavy, heavy fuel.
Mark Knopfler