I believe it's derived from hippo (horse) drome (arena) where horse races used to take part. It's no more common to use as a theatre or arena of some kind
Brighton Hippodrome was created in 1897.
Brighton Hippodrome ended in 2007.
Aston Hippodrome was created in 1908.
Bristol Hippodrome was created in 1912.
Beirut Hippodrome was created in 1885.
Birmingham Hippodrome was created in 1895.
Yes, Constantine did build the Hippodrome of Constantinople.
250 people can sit in the hippodrome at once
Queen's Park Hippodrome was created in 1904.
Adana Yeşiloba Hippodrome was created in 1962.
Royal Hippodrome Theatre was created in 1883.
Central Moscow Hippodrome was created in 1834.