It should be provable that thermodynamics guarantees that it will be more power-efficient to generate the light directly with electricity instead of using electricity to cool the room, then exploiting a thermal gradient with even a lossless Peltier element.
Assuming that T,H, and L are fixed, you're going to use more energy with a less-efficient process. So, you'll 'spend' more energy to get there.
If you're trying to run the process at fixed energy, you're going to wind up accepting a slightly higher T. If that's the case, you'd be able to use even less energy by powering the LED directly and moving the setpoint for T up to that same temperature.
