Your question made me think of a story Herodotus (5th c. BC) tells about the Egyptian Pharaoh Amasis:

his ordering of affairs was thus:--In the early morning, and until the time of the filling of the market he did with a good will the business which was brought before him; but after this he passed the time in drinking and in jesting at his boon-companions, and was frivolous and playful. And his friends being troubled at it admonished him in some such words as these: "O king, thou dost not rightly govern thyself in thus letting thyself descend to behaviour so trifling; for thou oughtest rather to have been sitting throughout the day stately upon a stately throne and administering thy business; and so the Egyptians would have been assured that they were ruled by a great man, and thou wouldest have had a better report: but as it is, thou art acting by no means in a kingly fashion." And he answered them thus: "They who have bows stretch them at such time as they wish to use them, and when they have finished using them they loose them again; for if they were stretched tight always they would break, so that the men would not be able to use them when they needed them. So also is the state of man: if he should always be in earnest and not relax himself for sport at the due time, he would either go mad or be struck with stupor before he was aware; and knowing this well, I distribute a portion of the time to each of the two ways of living." Thus he replied to his friends.

The story is fictional, bit it does demonstrate an awareness 'burnout,' as it were. I cant speak to other leaders, but actual Greek kings of the Hellenistic period prevented burnout by acting very much like Amasis, working hardest in the morning, then tapering gradually towards an evening banquet. Mornings were for administrative work: reading reports, writing letters, perhaps meeting with his closest advisers. Many Hellenistic palaces have suites of rooms right off the western side of a large central courtyard. These rooms would have received strong morning light, and were probably where the king worked in the morning.

The afternoon was when the king would hold court, hearing petitions and receiving envoys. The king was expected to be a public figure to whom his subjects could turn to settle disputes and right wrongs. King Demetrius was once supposedly accosted in the street by an old woman who demanded he stop and address her grievances. "I have no time," Demetrius told her, to which she responded, "then don't be king!" (Plutarch, Life of Demetrius 42). The story, once more, is probably fictional. But it does show rather nicely the theory behind Hellenistic kingship.

The day was usually capped off with a banquet. Once more, the king as a public figure was expected to eat, if not in public, then at least with a large group of people. The precise number of diners varied, but twenty or more was probably the norm. The evening feasts were also an occasion for work: they provided a relaxed atmosphere in which people could approach the king more freely. That being said, Hellenistic royal feasting followed the traditions of old Macedonian feasts, which meant raucous drinking, copious amounts of food, dancing, live entertainment, and the occasional deadly brawl. So a good mix of work and pleasure.

Naturally, kings varied this "typical" day with other activities that might have relieved stress (or at least made life less monotonous). Hunting was an extremely popular royal pursuit, especially in the generations right after Alexander the Great's death. Kings (and especially young kings) were expected to be warriors who led their men from the front in battle. As a result, some time must have been set aside for training, horse riding, etc. And kings might have other hobbies. Some wrote histories or penned works on generalship. Others delighted in taking cruises in their pleasure boats. Some just liked going to the public bath. As long as the normal cycle of business and audiences occurred frequently enough, kings could make a bit of time for other pursuits.