Does the temperature of a processor influence its energy consumption?
A CMOS chip ideally acts as you suggest with the dynamic power consumption proportional to the switching speed and the square of the power supply voltage (and the static power consumption with clock halted zero). 

However, as you try to lower the supply voltage (for reasons obvious from the above) the transistors no longer quite turn off all the way, and an ever larger static consumption appears. This is called subthreshold power leakage and increases with temperature. There is also gate oxide leakage.  

There are mitigation techniques, including circuit design and using exotic materials such as high-K dielectrics that can reduce the effect. At one time it was predicted that static power consumption could approach dynamic power consumption but I don't think that has happened.