I've picked up a couple of these 18650 boxes as well; they're great!
I'm wondering if I can do the following: (plan to wire it up and test possibly today)
Use the power box along with a 1f 5V super-cap...maybe a diode to keep the cap from feeding back into the power-box.
Have the PICAXE go into low-power sleep mode...wake up, take a quick temp reading, and go back to sleep...(all powered from the super-cap)...then, (after I determine how many times the super-cap can support that)...let's say for example, 5 times, on the 6th wake-up...fire up some LEDs to create enough current draw to trigger the power-box to turn on....super-cap recharges, and go back to sleep...wash-rinse-repeat...It would be fun to drop a picaxe off somewhere for a month of temperature readings or something, and power it all off of this nifty little box!