Many wrong answers, real reason is simple: multimeter was never supposed to measure spiky current correctly. There's a reason why it says DC.
For measuring sleep time leakage (which is pretty much DC), it's OK, but that 80mA peak could be actually anything. It's a complex waveform, you need the integral (or average).
You really need a scope, and a device such as Dave's µCurrent, or your own similar shunt amplifier circuit. You may be just barely be able to it with just scope, with relatively high-value shunt resistor.
As others have said, 5 seconds just to send temperature is massive waste of time. Even if the current actually isn't 80mA, it's still something quite high, so fix this and it'll last even longer. You should be able to do this in 50ms easily.