Yes, but it’s not a timed response as much as a temperature response (and admittedly, the cold always comes after the warmth, from the leaves point of view).

We know that carotenoids are present all year round and are seen in autumn because of the chlorophylls degradation. In contrast, anthocyanins are mostly produced in autumn, in response to cold stress. This could be because these pigments have a protecting effect against cold.

About carotenoid synthesis.

About anthocyanins and cold stress.