This can't be serious....
It's a peltier device in a fancy package and hyped marketing wank.
It will heat/cool a tiny area of skin behind the neck, perhaps helped by a fan.
Given the size, for it to last 24 hours, it isn't going to produce much cooling or heating.
Reason being, peltier devices are inefficient and the battery won't have a lot of capacity.
It also appears to have a fan (reference: holes in device seen through special shirt and text indicating "air volume" in Gizmodo article).
The primary function of the fan would be to get rid of the excess heat/cold on the opposite side of the device (the part not touching the skin).
Peltier devices need a way to remove heat from the hot side or else the cool side won't stay cool. The temperature differential must be less than 70 C or so to be at all effective.
It is not clear if the fan also spreads heat/cold to the user wearing the device as well. From the video, it may.
Let's get into engineering specifics:
According to Sony, the Reon weighs 85 grams..
Let's assume that the battery is 80% of that weight, or 68 grams.
Let's assume the battery is lithium ion. I cannot read Japanese so I cannot confirm, but what else would it be?
Let's assume the other electronics inside the device (the wireless connectivity, temperature monitoring, battery voltage monitoring, etc.) are all negligible power draw. It is not true of course, but it gives us simpler calculations.
Lithium ion batteries have a range of 100-265 W-h/Kg energy density, so let's use 200 W-h/Kg as a middle number.
68 grams is 0.068 Kg, so the battery has 0.068 Kg x 200 W-h/Kg = 13.6 W-h of energy.
Sony says the device will run for 24 hours on a single charge, so power draw per hour is 13.6 W-h / 24 hours = 0.57 W
The lithium battery will likely be 3.7 V nominal, so average current draw will be 0.57 W / 3.7 V = 150 mA
150 mA certainly seems like a reasonable current for a battery of this size, so I think these calculations are correct.
The battery would need to have a total capacity of 150 mA x 24 hours = 3600 mAh. This is a perfectly reasonable battery capacity and is in fact a standard size readily available everywhere.
Looking on eBay, 3600 mAh flat lithium battery packs seem to weigh around 55 grams, which is under my estimate of 68 grams.
So the battery is only 65% of the total weight of the device. And it means that the battery Sony is using is actually 247 W-h/Kg in terms of energy density, which is near the high end of lithium technology, but still within the 100-265W-h/Kg range given by Wikipedia.
So, if the above calculations are correct, a peltier device running at 150 mA isn't going to do much in terms of heating or cooling.
But I suppose it will give many people a placebo effect. The back of the neck is a sensitive area.