Do the Hippodrome and Hagia Sophia fall under the pax romana or the Byzentine empire period of time?

The Byzantine Empire of course. Hagia Sophia ("holy wisdom") was a major christian edifice, and christianity hardly yet amounted to much in the Pax Romana. Btw ammianus got one thing wrong: 180 CE marked the end of the age of the Antonines, or Good Emperors, NOT the end of the Pax Romana. In fact, the reign of Commodus, from 180-192, was far more peaceful and stable than the tumultuous years of Marcus Aurelius's reign (161-180). And while there were civil wars in the period 193-97, generally there was stability until the end of Alexander Severus's reign in 235 CE. (Historians don't say the pax romana because of the civil wars of 68-69). Serious internal strife was the exception down to 235; it became the rule only beginning around then.