If it were a plain LD like a red or bluray this is something that can be done.
But, when its a DPSS laser, there are a few things to consider. Cooling can lower the wavelength of the pump diode as heating can raise it.
You want as close to 808nm going into the crystals, not 807nm not 809nm. Changing the temp from the ideal which should be around 20-22C, will begin to lower your output of green as you saw when the module got warm. The wavelength of the LD begins to rise, and the efficiency of the crystals drop resulting in lower mW of 532.
Now when you cool the entire body, the crystal set also drops its temp, this is where the mode switching becomes a problem, when its too cold.
There are heaters in more elaborate systems for this reason..
Try for a decent room temperature, and you should have things back to normal. I keep my large heatsinked greenie portables in a nice warm cozy pocket in the winter months for this very reason.