The OEM batteries in Tundras is of poor quality. Mine was oozing white stuff the first year of ownership. I bought an Optima Red Top.
I was in the -40C vicinity here last week as well. Global warming is B.S.

Mine was not plugged in, but I have synthetic oil and the the Optima. I find that by turning the key on for a couple seconds before cranking it seems to help. I am guessing this gets a little pressure in front of the injectors, hence getting fuel into the combustion chamber a little quicker. The truck started quickly and relatively easily all week, not plugged in once.
