Almost any GNSS receiver is by design power hungry. Modern mostly SDR constructions have faster time to (reasonable) fix, but still it involves large amounts of essentially opportunistic computantion to acquire and track the CDMA downlinks. This is one of the arguments why Spy-Fi style inconspicuous and globally accurate and useful trackers are nonsense. The whole process of associating with 2G GSM network and sending one MO-SM message is incredibly energy expensive, but getting reasonable GNSS fix is in completely different barpark.
Possibly more feasible if you periodically capture and store a few ms raw IQ, then transfer a quick burst when the handler comes near, to do the nav solution offline.
