Is a cool roof appropriate in a climate that experiences cool or cold seasons?
Yes, unless the climate is cool or cold all year. This is true for buildings whether or not they have mechanical heating and cooling.
In a cool or cold season, a building with a cool roof will be slightly cooler than a building without one, which may increase discomfort or require additional heating. This is called the "winter penalty". However, the benefits provided by a cool roof in warm or hot seasons outweigh the 'winter penalty'. This is because in cool or cold seasons, the sun is in the sky for less time and at lower angles, so it does not have as much of an effect on the temperature inside the building. Therefore, neither does the cool roof.