The idea that it was a different time in different places predates the empire. The Greeks understood the spherical nature of the Earth since at least the fourth century BCE, with Eratosthenes calculating a good rough circumference for the Earth. Hipparchus of Nicaea in the second century BCE even proposed a method to measure the differences, record local times during eclipses.

I'm not aware that anyone actually making an attempt to measure time difference. But there are later references indicating that at least the educated classes were aware that noon would take place in different places at different times. For example Pliny the Elder makes reference to a 24 hour time cycle in his Natural History.