Defining lunar time is not simple. Although the definition of the second is the same everywhere, the general theory of relativity dictates that clocks tick slower in stronger gravitational fields. The Moon’s gravitational pull is weaker than Earth’s, meaning that, to an observer on Earth, a lunar clock would run faster than an Earth one. Gramling estimates that a lunar clock would gain about 56 microseconds over 24 hours. Compared with one on Earth, a clock’s speed would also subtly change depending on its position on the lunar surface, because of the Moon’s rotation, says Tavella. “This is a paradise for experts in relativity, because you have to take into account so many things,” she adds.

Not only do you need leap seconds to keep solar time and atomic time in sync, you’ll need a different kind of leap second to keep Lunar atomic time and Earth atomic time in sync, because mass distorts spacetime.

See some links in reference.

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Reference:

1. What time is it on the Moon?
2. Time is?
3. NASA Moon Mission Set to Break Record in Navigation Signal Test
4. New Lunar Satellites Will Enable Autonomous Space Travel to the Moon for Astronauts
5. A Case Study Analysis for Designing a Lunar Navigation Satellite System with Time-Transfer from Earth-GPS

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