I've had this issue on 4 occasions, since having several laptops since 2007. It didn't matter what brand(Toshiba, HP, Dell, ASUS)
Everything you're talking about is plausible.
My situation was resolved in the following way. As much as I have taken on board ALL the comments that OTHER people have also suggested, realistically all the 4 occasions were resolved after finding the one factor in all of them. When I replaced a NEW battery, the overheating issue wasn't a factor anymore. When I returned the OLD battery, YES I had what everybody complained about. In all 3 occasions I did the following; while assuming the battery was OK because I had often kept the AC connection plugged in, of course seeing the FULL BATTERY image made me feel the BATTERY was fine. It was easy to just go on with whatever I was doing. When I came across the issue with the hard drive overheating, I just happened to have a friend that owns a computer store that gave me a NEW or currently working battery(he tested for as reliable), then plugged that into my laptop. For THAT TEMPORARILY REPLACED battery, the hard drive OVERHEATING issue was pretty much non existent. Curiously, I replaced MY OLD battery that I ASSUMED was fine(because I LOVED to see it FULLY CHARGED), then I pulled out the AC adaptor to allow the laptop to run off the battery. NOW THIS is something that I OBSERVED in ALL 3 occasions, that allowed me to now SHARE this observation with you. In ALL 3 occasions, the battery RAN DOWN within the first 20 minutes(none of them went longer than 20minutes(ie 4minutes, 12minutes, 15min, 20min)) and the battery alert came up 7minutes remaining your battery is low. Basically the none of the batteries lasted as they would usually run for at least 2 hours.
YES I also cleaned out the fan and blew out the motherboard of any dust. My friend has the laptop repair shop. Even keeping the OLD battery, the same issue would arise, the Hard Drive would overheat, while I had the AC connector plugged in, AND if I took out the AC connector, the battery did not last very long(in my case under 20minutes).
Therefore, the issue was the battery that could not hold it's charge anymore. I had been keeping the battery plugged into the AC for about 3 years ongoing, & yes I also occasionally ran it on it's own without the AC while sitting on the bus or in the airport. BUT I MOSTLY kept it connected on the AC, like ALOT of people, FEELING the ASSUMED comfort that I was always 100% full power when something went wrong(blackout poweroutage in asia Vietnam, Thailand, Cambodia). In the past I INITIALLY did what everyone else seem's to always bring up, clean the fan area, blow off the dust, check power settings, blame the hard drive. After REPLACING the battery, obviously because in all 3 occasions it could not CONTINUE past 20 minutes without the warning LOW BATTERY POWER. In my situation, which SOUNDS like what everyone is talking about, the battery was the issue.
Is the hard drive LESS HOT when only on the BATTERY(not connected to the AC adaptor), YES in my case it was LESS HOT, whether it was on the NEW BATTERY, or even on the FAULTY BATTERY(that did not last longer than 20 minutes with the warning LOW BATTERY POWER).
But in ALL occasions, when the FAULTY BATTERY(assumed to be working well with FULL POWER STORED) was connected to the AC connector, the Hard Drive and generally underneath the LAPTOP area was OBVIOUSLY very HOT, then often would cause an automatic shutdown after some time(within 3 hours sometimes or after many hours under 48 hours). It didn't matter whether I had the AIRCON in the hotel room or a fan blowing directly on the laptop. The Faulty battery always appeared to have the issue with the UNDERNEATH feeling GENERALLY UNCOMFORTABLY HOT(hard drive area, cpu, memory chips).
Replace the battery after seeing it not last very long like I did, then YES for the next 2-3years we MAY NOT have the same issues again(Max 4years). BUT I now know the symptoms of this after 4 occasions. I don't waste time, after watching the battery not last long on it's own. I just get a new battery.
Will there be times that your AC ADAPTOR is at fault? YES, but in that case, your battery won't be charged fully, and you will see that it doesn't appear to be CHARGING the battery or show connected to the AC power. Replace that too in that case. Only cost $6-$8 in Asia, and lasted LONGER than the batteries. THose occasions I didn't have overheating. I just didn't see the battery being charged.
Hope this HELPS EVERYONE.