At each 90 degree jump you might, say, put in one unit of work to move two units of heat. The problem is at step two you have three units of heat, two of which

At each 90 degree jump you might, say, put in one unit of work to move two units of heat. The problem is at step two you have three units of heat, two of which came from ambient heat. At step three you have 4.5 units of heat, two of which came from ambient heat. Step five there are 6.75..and so on. You're expending a lot of energy moving energy you put in so you get diminishing returns. The theoretical limit is T_hot / T_change (in absolute units like kelvin). If your change is roughly the hot temperature, there's no benefit.