"At the time" versus "at that time"

They are both correct, but "at that time" has that stilted, official-ese sound that some government employees like to affect when they are talking to the press. "At the time" is what normal people say.

Exception: if you are distinguishing between two times, especially between some former time and now, use "that". E.g. (emphasis added) "The National Association for the Advancement of Colored People was established in 1909; at that time, 'colored people' was considered the polite phrase for 'black'."

at "that" time is usually used for sudden events. For example: at "that" precise moment police have arrived), where as at "the" time is used for events that happened long ago. For example: at "the" time of his birth...

at that time is used about a time in the historic past:

In Celtic Ireland women were allowed to own property, which was unusual at that time.

at the time is used about a particular occasion:

I didn't have any money on me at the time.