The viscosity of water increases at lower temperature, and the submarine would have more trouble moving through it.

Viscosity almost doubles going from 70 F to 32 F.

There are confusing factors. Water density and viscosity at low temperatures would not significantly favour higher speeds, and might even limit the operational speed, but the low vapour pressure of cold water outweighs that. One consequence of high-speed underwater motion is cavitation, the formation of bubbles of vapour and dissolved gases. Cavitation occurs wherever there is drastic pressure reduction in water, and it is very difficult to move water fast without creating zones of reduced pressure. In cold water the solubility of gases is higher and the vapour pressure lower, so one can achieve a higher speed before cavitation begins and a still higher speed before it becomes a major limitation to the speed of craft of any given design. Such zones obviously occur behind the screws, but cavitation in the slipstream and the vortices of the wake can trouble submarines even more. Also, increased viscosity discourages vortex formation.

In military practice cavitation is a critical limitation on underwater speed, less because of propulsion inefficiency than because cavitation bubbles are conspicuous sonar targets. Submarines on the prowl must keep well below top speed to avoid giveaway cavitation that might betray their position, and the reduced tendency to cavitation noise or echoing bubbles would favour higher tactical speeds in colder water.

Perhaps it's to save on heating bills by not spending any more time than necessary in the cold conditions.

One Word. Shrinkage.

At the depths that submarines operate at the water temperature is effectively the same anywhere in the world as the sun cannot penetrate to this depth to warm it.

A submarine is therefore unlikely to experience a large enough variation in temperatures to make any difference to it's speed.

I think you misheard.