From https://wiki.openstreetmap.org/wiki/DE:Tag:leisure%3Dhorse_riding:

For a  riding arena use leisure=pitch + sport=equestrian
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The tag leisure=horse_riding is used to tag an equestrian facility, often called equestrian centre, riding centre or riding stables, where people practise horse riding, typically in their spare time. The tag is for the whole area which may contain one or more riding arenas, stables, riding halls, other related buildings or paddocks (compare also Riding).
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Use this tag only once for the whole facility, don't use it multiple times on different objects belonging to it (one feature, one OSM element rule