You should record the event time separately [1] from the beginning/end of the measurement period - the first as a time instant in UTC ('wall clock'), the latter as a difference between two values of a guaranteed monotonically increasing clock.
This is something you need to do regardless of leap seconds to handle things like NTP kicking in to adjust your local wall clock time.
[1] - Depending on the programming environment, these might be either different timestamp types recorded separately or converted appropriately to reflect different uses, or a smart combination type like time.Time in Go.