Русско-англо-русский юридический словарь

“At the time” is the more common choice used by most, if not all, native speakers. It simply means that there’s a time when an event happened, and there isn’t anything that can be done to influence that event anymore because it happened in the past. (Grammarhow)

Generally, “at that time” is more sudden or more specific. We use it when we want to emphasize the exact time when an event might have occurred, and it’s especially impactful if someone has already asked us about one or two times, and we specify which one applies. (Grammarhow)