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...
Don't try this at home.