Showing intellectual and artistic promise, Coste began his studies in the studio of Michel-Robert Penchaud, architect of the département and the municipalité.

His time in Paris was a pivotal one in his life—there he met the geographer Edme-François Jomard, who put him in touch with the viceroy of Egypt, Muhammad Ali Pasha, who took Coste as his personal architect in 1817.

In Iran, Coste and the painter Eugène Flandin were authorized to visit Iranian Azerbaijan, Isfahan, Shiraz and the ruins of Ecbatana, Bistun, Taq-e Bostan, Kangavar, Pasargadae and Persepolis, where he made many sketches.

He continued via Nineveh, to which the archaeologist Paul-Émile Botta was also travelling to begin his excavations.His Middle East journey aroused the interest of Louis Philippe I and gained Coste the post of chief architect of Marseille in 1844.

Coste was also the originator of two other architectural projects in Marseille—the construction of the faculté aux allées de Meilhan, and a museum with château d'eau at Longchamps.