As 'kings in medieval times' is an incredibly broad descriptor, I will restrict myself to elite recreation in the Carolingian kingdoms.

Judging from contemporary annalistic sources, the most common leisurely activity for Frankish aristocrats during the 8th and 9th century was hunting. Most Carolingian rulers are mentioned as making arrangements to hunt during their stay at various royal estates, particularly during autumn. A specific villa at Orville is, for example, named as a favourite hunting destination for Charles 'the Bald'. The use of hunting dogs and falcons are also frequently mentioned in this context.

But hunting was far from safe for nobles; King Louis 'the German' is noted for falling off his horse after following a deer into a thicket in 865, and Aistulf, King of the Lombards, died upon falling from his horse a century before, in 756. There are other examples, but these are the first to spring to mind.

Sources:

Bernhard Scholz, trans. Carolingian Chronicles: Royal Frankish Annals and Nithard's Histories (Ann Arbor, 1972).

Janet Nelson, trans. Annals of St. Bertin (Manchester, 1991).

The term "spare time" is a bit misleading, there was not really a difference between private and public life for a king.

As it has been pointed out by other responders, hunting was a popular "hobby" for the nobility. But think of who was with the ruler at this time: other noblemen who had interests and would discuss them during the hunt. Furthermore, the king could also proof his martial ability. As u/Funkraum pointed out, hunting was not entirely without danger, so one can wonder why they would risk their skin for such a thing, and again the answer lies in the symbolic expression of prowess. An anecdote from the 16th century: The duke of Wurttemberg personally killed a particularly large wild boar that roamed his lands (this species of boar is today extinct in western Europe). This included standing with a spear against the attacking animal (think Robert Baratheon in ASOIAF). The duke had the beast stuffed, mounted on wheels and hidden in a cabinet in his grand hall. A secret lever would make the boar come out of the cabinet and scare the bejeezus out of the visitors, and proving over and over again what he stood against (Source: Page of the Museum http://www.schloss-urach.de/wissenswert-amuesant/anekdoten/wildsau/).

Other "leisure" activities could of course include chess, there's a reason it is called the game of kings. The 12th century Lewis Chessmen, today in the Scottish National Museum in Edinburgh, are of walrus ivory and the craftsmanship makes them fit for a high nobleman (or noblewoman!), or even a king. (http://www.nms.ac.uk/explore/collections-stories/scottish-history-and-archaeology/lewis-chessmen/). But again, who did they play it with? With their peers with whom they made politics and chummed up to.

Other examples I can think of are for example Falconry (also mentioned above): Emperor Frederic II even wrote one of the standard works about hunting with birds De arte venandi cum avibus (Scan by the Vatican Library: http://digi.vatlib.it/view/bav_pal_lat_1071). With this particular monarch, nicknamed stupor mundi 'marveler of the world', you can actually add writing, and science to the list.

John Lackland - then still a prince - received a valuable bird, named Tabur, as a gift from Richard Fitz Nigel, the bishop of London, (Ralph de Diceto, ii: Ymagines Historiarum Rolls Series, p. 102). So this is an actual example of how leisure and work (i.e. political influence and friendship through a gift) was basically identical in medieval politics.

Lastly: architecture: The aforementioned Frederic II built the representative Castel del Monte, until today a marvel of architecture as a hunting palace (yes, hunting again). Richard I Lionheart planned the famous Chateau Gaillard, which included the newest technology in siege warfare of his his time. And in France you have the example of the Sainte Chapelle - I believe by Louis the Pious Saint Louis (thanks u/CubicZircon) - fashioned after the king's taste. Once more, theses buildings were of course something these rulers invested into heavily and were really personally involved - but ultimately they were buildings symbolizing their power and might. The kings did not have leisure time, such a concept is simply not 'medieval'.

How much spare time would a medieval king even have? Modern heads of state don't seem to get much at all, and that's with private jets and satellite communications to drag them back into business at a moments notice. Is it at all comparable? Could a king afford to be away from his governing apparatus, if you will?

Your question is a bit tricky to answer because the duties of kings differed drastically over such a large timeframe and because it is unclear if by spare time you mean purely leisurely and recreational activities. Thus, this might not answer your question if you are curious what medieval monarchs did to escape the responsibilities of rule.

In the early middle ages, rulers are often described as itinerant kings, meaning that they traveled a lot - an effective king wouldn't rule from the 'great hall' alone. Feudal kingdoms were decentralized, and a king's authority was only exercised when it was enforced. The political structure of the time forced kings to move throughout their domain to rule effectively. Itinerant kingship is most well documented for the early Frankish and German rulers, especially the Carolingians. In this time, it was important for kings to demonstrate their personal power. It also legitimized the rule of a new king to travel throughout the country and have the local nobles pay fealty.

For one example, due to this nature of early authority, and because early medieval Europe was often lacking in infrastructure, the itinerant Ottonian and Salian kings came to rely upon monasteries to feed and accommodate the royal court as it traveled throughout Germany.

If you ask only about leisure, many kings, such as Charlemagne, would go hunting. Other popular relaxations were palatial baths or hot springs, both of which were parts of Charlemagne's palace at Aachen. He would often invite nobles, friends, and bodyguards to bathe with him.

Sources:

Bernhardt, John W. Itinerant Kingship and Royal Monasteries in Early Medieval Germany: C. 936-1075. Cambridge: Cambridge University Press, 1993. Print.

Einhard: "The Life of Charlemagne", translated by Samuel Epes Turner (New York: Harper & Brothers, 1880). http://www.fordham.edu/halsall/basis/einhard.html