Suppress the urge to avoid cleaning teeth that are sensitive to cold! Teeth that have suffered damage from gum disease will always be more sensitive to cold. Avoiding them only makes it worse.
After any dental treatment, teeth may become sensitive. This is their way of letting you know that they've been injured. Any injury (cavity, tooth clenching/grinding, gum infection) can injure the nerves in a tooth. This should not last long if the teeth are kept clean. If the teeth are not kept clean, the sensitivity will remain or get worse.
If your teeth are especially sensitive, consult with your dentist. This could be a sign of the need for root canal treatment, a gum tissue graft, a desensitizing toothpaste (with potassium nitrate), or a concentrated fluoride gel (0.4% stannous- or 1.1% sodium-fluoride) may do the trick.