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
Bristol Hippodrome was created in 1912.
Beirut Hippodrome was created in 1885.
Brighton Hippodrome was created in 1897.
Brighton Hippodrome ended in 2007.
Aston Hippodrome was created in 1908.
Birmingham Hippodrome was created in 1895.
Yes, Constantine did build the Hippodrome of Constantinople.
Royal Hippodrome Theatre was created in 1883.
Hippodrome Theatre - Baltimore - was created in 1835.
Central Moscow Hippodrome was created in 1834.
Adana Yeşiloba Hippodrome was created in 1962.
Queen's Park Hippodrome was created in 1904.