I had the exact same thing happen to my A7C. I was in video mode and tweaking settings. Eg I was going back and forth between live view and menus. No recording, mind, and at the time the heat setting was set to normal, not high.
Camera all of a sudden gave high heat warning. I live in Singapore, so it is daaaang hot here, so that explains it happening to me. What I am getting at is that if you are in Video mode the video processing is going on, generating heat, even when in the menus.
If you change from video to camera mode it does not shut down (in my experience), so this "proves" (with a sample of 1) that it is related to the ongoing video processing and whatever setting you have set for video.
Changing the heat setting to "High" fixed it in all situations for me.