Human Beings don't retain some level of consciousness when asleep (offline). So why would the Doctor? We may retain a few minutes of a particularly vivid dream but that's it. The Doctor would log the time of his reactivation and probably compare it to his last time of deactivation. Noting a passage of time between the two records. But that's the same as us checking the clock when we wake up.
Perhaps when his holographic projection in Sickbay is terminated his program is transferred to some type of Moriarty-like box down in the computer core. Where he spends his time in a virtual environment doing whatever he wants to. Maybe he has a ice rink in there where he's learning the winning routine from the last Olympic figure skating competition.
Perhaps when his holographic projection in Sickbay is terminated his program is transferred to some type of Moriarty-like box down in the computer core. Where he spends his time in a virtual environment doing whatever he wants to. Maybe he has a ice rink in there where he's learning the winning routine from the last Olympic figure skating competition.