Maybe it's possible to recycle the processor's power by using the heat to electricity way like those ventilators they make for wood stoves! And so lowering the power the CPU uses!
I like that style! Your idea would be novel - except it causes heartburn. You are talking about embedding a thermocouple or other form of Peltier device inline with the CPU heat flow to extract energy. That comes at a cost.
Any energy conversion requires the same idea as a "voltage drop" and "current flowing through it". Heat is the same.
Let's start with thinking about motors. In motors the output mechanical power is P = torque x speed, which requires the duality on the electrical side of "current" (very strongly related to torque in the motor) and "voltage (very strongly related to speed). AC motors can be a little more complex than that, but DC motors are on a first order, exactly like that. Speed correlates to motor voltage, torque relates to motor current.
When a Peltier device is generating electrical power, it is creating an output voltage. And if there is a load, some output current. The thermal part of a Peltier device will have temperature drop across it (think of thermal resistance) as the heat flux goes through it (like electrical current). So heat flux (Q) on the thermal side is related to voltage on the electrical side, and temperature drop on the thermal side (delta T) is related to current on the electrical side.
On the aerospace projects that I've used Peltier devices on, we even used this to modulate the heat flow. Short the electrical side together (no voltage drop) will give you the lowest temperature drop on the thermal side. Opening the electric circuit side gives you the largest temperature drop on the thermal interface part.
But note: present day Peltier junctions are really pretty bad, and are not efficient. For instance, they would not terrible to cool your house, as they exist right now.
But there have been CPU coolers made that you put electrical energy into a Peltier junction to cool the CPU more. Just like a heat pump on your house.