This is the Odeon near the State Agora.

From the Wikipedia: "Odeon (from the Ancient Greek ᾨδεῖον, Ōideion, literally "singing place", or "building for musical competitions"; from the verb ἀείδω, aeidō, "I sing", which is also the root of ᾠδή, ōdē, "ode", and of ἀοιδός, aoidos, "singer") is the name for several ancient Greek and Roman buildings built for music: singing exercises, musical shows, poetry competitions, and the like."

Its roof was probably made of wood.