Predation risk in the wild varies across the day in a relatively predictable way, as a function of environmental conditions, such as light intensity and temperature, and of predator habits. Prey animals can thus adjust their own activity rhythm to avoid the most dangerous periods. We studied this situation in a coevolutionary perspective, considering that, if prey spread their activity across the day to counter predator temporal preferences, these preferences may in turn change to track prey activity. We therefore built a game-theoretical model to derive evolutionarily stable activity patterns for a predator constantly trying to maximize its chance of capture and a prey trying to minimize this probability. Key parameters concern circadian variations in environmental conditions and their influence on predator hunting efficiency and energy demands of the prey, which dictate its total amount of activity. The model predicts high levels of prey activity during periods of reduced predator efficiency. The predator may then either track these activity peaks and copy the activity patterns of its prey, or concentrate its hunting effort during periods when prey activity is low but conditions favour hunting. In the first case, predator and prey activity patterns will be synchronized. In the second one, they will exhibit strong temporal segregation. We show how these diverging scenarios emerge and how they can help to disentangle the wide variety of situations existing in the wild.