The short answer for most cooling blankets is zero electricity required. But the category includes several different types of products that work very differently. Here is the full picture.
Fabric Cooling Blankets - Zero Electricity
Standard cooling blankets made from nylon or cooling fiber use no electricity at all. They work through passive thermal conductivity, the physical properties of the fabric material draw heat away from your skin without any power source. There is nothing to plug in, nothing to charge, no pump, no compressor. You wash them, air dry them, and sleep under them. The only cost is the purchase price.
The Stillwell Cloud uses zero electricity. See The Cloud here.
Water-Cooled Blankets - Moderate Electricity Use
Water-cooled systems use a pump that runs continuously at typically 50-150 watts. Running this for 8 hours per night adds roughly $5-15 per month to your electricity bill depending on your local rate. Over a year this is $60-180 in running costs on top of the $400-800 purchase price.
Electric Cooling Pads
Some cooling mattress pads use embedded cooling elements powered by electricity. Power usage varies by system, typically 10-60 watts. Less than a water-cooled system but still an ongoing cost that accumulates over time.
The Case for Passive Cooling
A fabric cooling blanket with no running costs, no setup, no noise, and no maintenance solves the problem for the vast majority of hot sleepers at a fraction of the total cost of active systems.