So I contacted Sony’s tech support, and they told me both that shutting down in 30 minutes at room temperature is perfectly normal, and that if I sent it back to an authorized repair center, that they might be able to fix it. Wait, is it normal or does it need fixing? The tech support person seemed to think it was both.
That’s the last straw for me. I’m sending it back. According to DC Rainmaker neither the GoPro Hero 5 Black nor the Garmin VIRB Ultra 30 have this problem. I think I’d rather have less than perfect image stabilization but be able to use the whole battery life than have great image stabilization but be unable to film more than 40 minutes of a race.