Yes, Constantine did build the Hippodrome of Constantinople.
Constantinople
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.
Bristol Hippodrome was created in 1912.
Constantinople
The first Hippodrome in today's Istanbul was built by the Roman emperor Septimius Severus in the beginning of the 3rd century as one of series of reconstructions in ancient Byzantium. In fact, he was the one who had punished Byzantium and destroyed almost the entire city for supporting his enemy, clearly the loosing side. The hippodrome was enlarged and renovated by Constantine the Great when the city was made the capital of the eastern Roman Empire as Constantinople in the 4th century. Some claims that the capacity of the hippodrome reached 100,000 seats, but recent findings show that this figure appears to be exaggerated.
Hippodrome Theatre - Baltimore - was created in 1835.
Central Moscow Hippodrome was created in 1834.
Adana Yeşiloba Hippodrome was created in 1962.