Even the stackoverflow answers say as much. Most of the thermal ramping occurs in the first five minutes and then stabilizes after cache is exhausted. As all these dslr’s allow you to record 29m50s videos back to back all day long it’s also indicative that it’s not a big enough issue in returns to warrant addressing in their firmware.
As you’re not writing to the SD card, hdmi passthrough should yield an overall lower operating temperature after an hour than writing to the SD card when recording video after even 10 minutes.
I tried getting to the source of this rumor. Didn’t find one particular origin, just speculation amongst Redditors and Twitch users about the 30 minute limit and bypassing it with the Nikon or Canon firmware mods will void your warranty.
It will however introduce more noise. Particularly at higher ISO’s. Astronomy photographers (a major driver of the firmware hacks in the first place - see “Nikon stareater bypass”) have cooling solutions available there too.