"Temporal observatory" was a term used by Temporal Agent Daniels to describe the small, hand-held device which he used to monitor the timeline. It operated by projecting a holographic representation of the timestream, allowing Daniels to supervise various activities being carried out by time travelers.
In 2151, Silik attempted to steal the temporal observatory and deliver it to the Suliban Cabal's mysterious benefactor, which would have provided his faction an advantage in the Temporal Cold War. Captain Archer destroyed the observatory with a phase pistol rather than let it fall into the wrong hands. (ENT: "Cold Front")