OK, let's run the numbers. I'll just do cooling, heating would be comparable. Also, I'm going to be generous and assume optimum performance from all the elements of the system.
So, they don't say it, but this is obviously a peltier device. It has a 400 mWh Li-Ion battery so 0.4 * 3.7 = 1.48 Wh. 1.48 Wh * 3600 seconds = 5328 joules. Peltier devices can, in ideal circumstances, shift more heat than energy consumed. Let's assume an ideal CoP of 3, so this thing can, again under ideal circumstances, shift 15984 joules across the peltier device before the battery runs out.
The specific heat of fusion of ice is 333.55 j/g, so the heat shifted by this device is equivalent to the work done by melting 47.92 grams of ice at 0C to produce water at 0C. So, it'll do the cooling work of, let's be generous, 2 ice cubes, one held on each wrist.
This, any arts graduates listening, is the power of basic (I mean really basic) science to keep your money in your wallet when shysters come calling with gadgets on Kickstarter.