Micro Watt Meter in Extreme Low Temperature
Datalogging etc is probably what you want to do, but if you wanted no software, or extreme low power, this is an option:

If you measured current after the regulator, or at a fairly constant voltage point, then you would simply need to integrate current.

This can be done without using a microprocessor using an integrator type voltage-freq convertor, and a counter. In the extreme case using micropower opamp and comparator below, it will draw less than 1uA, while running continuously. 







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BTW, because this converter is truly integrating the current in C1, it can accurate measure the current drain of intermittant loads that are sleeping and waking. 

To directly integrate power:


R5 can be made inverse proportional to voltage (R~1/V), then it becomes a multiplying power meter. A pwm of R5 with duty cycle 1/V can do this.
If REF can be 1/V (over the limited working range) then it will integrate power. Note that V is effectively sampled only when the integrator resets. [suggestions for a 1/V circuit welcome]


C1 polypropylene or polystyrene preferably. Certainly low leakage.
R2C2 delay ensures C1 can be reset. reset switch is low leakage.

PCF8593 I2C clock chip can also be configured as a counter, which you could read out at any later (warmer and better powered) time.

If you were using a micro, this is still a nice low power current integrating current sensor, which integrates the current even when the uP is asleep. If input leakage current is low enough, then the io pins can be used as a reset switch, and using a comparator keeps the cpu operations quick. So you only need R1,5,Amp1,C1. All analog functions can sleep during integration, so total current including th uP could be under 5uA







simulate this circuit