First off, you can't bring anything to 0K actually, as it is an unachievable temperature.
But, getting very close to it, at some point the propellant gas would become liquid, and even colder it would solidify.
You cannot get down to 0 K, okay ? Very best tech can reach pico-Kelvins, but even liquid nitrogen, which boils at 77 K (−195.8°C or −320.4°F) will have *interesting* effects.
Um, is this deodorant a 'pump action' version, or 'propellant' driven ? Does label warn against fire, likely an alkane mix ?? Or is it a rare (?) CO2 'bag in can' model ?
Whatever, the 'carrier' solvent will soon freeze solid. Any 'propellant' vapour will later freeze, too. If a plastic container, it may hit 'plastic transition temperature' somewhere below -20°C (-4°F) and become very, very brittle. If an aluminium alloy, it may become embrittled, but I don't know the metallurgy. Certainly, the pump / seal / valve assembly will suffer, may leak when thawed. Anything from 'fizz' to 'FWOOSH', potentially lethal if flammable...