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The problem is the voltage drop. The bigger voltage difference it has to handle, the hotter it gets as its the loss from "internal loss current x voltage drop" is converted into hat.

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That only applies to linear regulators.

The router on this page uses a switching regulator. Switching regulators have a relatively constant power draw over the entire input voltage range.

The heat has to come from the input power though. If I'm pumping 5W at 12V, or 5W at 20V into the device, with it idling, presumably the output from the devices voltage regulator circuit is the same voltage, and the downstream components are consuming the same wattage (let's call it 1A at 3.3V downstream, so 3.3W), the both the 12V and the 20V input would have 1.7W of heat-loss. The article shows that as voltage goes up, power actually drops, which would imply that it is producing less heat.

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