No, the Romans just generally did not wear hats, particularly the upper classes. Every Roman statue ever depicts a full head of hair, occasionally with veils, wigs or hairnets, but not hats. Apparently even Caesar's famous laurel wreath was meant primarily as a disguise for baldness.

https://en.wikipedia.org/wiki/Roman_hairstyles

Suspect. Laurel wreaths would make the world's worst toupee, as they are empty in the center, and provide the most coverage where balding men have the most hair.

Probably hats were associated with people who work in the sun like farmers or soldiers.. Togas were also impractical clothing compared to a tunic.

Every roman statue depicts politicians and leaders who lived in the city. Peasants, farmers did not get statues.