The behaviour of "frozen filament" will entirely depend from the specific formulation of it.
The term "flexible filament" encompasses a variety of different polymers as for example: thermoplastic elastomers like TPE and TPU (e.g.: ninjaflex), copolymers (e.g.: bendlay), copolyesters (e.g.: Ngen Flex), polycaprolactones (e.g.: PCL), etc...
Even in those broad classes of chemicals, the amount, type and quality of additives will affect the physical properties of the filament a lot. In fact rigid.ink even produces a flexible PLA that proves the point of additives radically affecting the properties of the main material.
In general, all materials lose elasticity at lower temperatures (a Space Shuttle came down because engineers failed to account for this). Polymers that do not contain water are unlikely to crystallise though, so I would expect it to become stiffer but not to fail catastrophically at 0°C.