Do BAD capacitors work less in the cold?
The capacitance of an electrolytic capacitor decreases slightly with temperature and ESR (Equivalent or Effective Series Resistance ) increases greatly. Bad electrolytic capacitors generally manifest by having high ESR rather than low capacitance, so I suspect this effect is what you are seeing. From Nichicon's manual (response of a good capacitor): 



ESR increase is as a result of the electrolyte drying out in the capacitor. The aluminum cans are sealed with a rubber bung at the bottom which is crimped into the case and penetrated by the leads. 

Another possibility is upward drift of the value of the startup resistor in the power supply. Higher temperatures mean higher gain for some semiconductors and resistors exposed to high voltage DC tend to drift.  

The annoying whining noise you are hearing can be mitigated by purchasing a new set- they are very cheap these days.