High-precision time synchronization in astronomy is applicable almost exclusively for applications like Very Long Baseline Interferometry (VLBI) with radio telescopes and for Imaging Atmospheric Cherenkov Telescopes (IACTs) observing extensive air showers from gamma-rays. IACTs require nanosecond-scale synchronization for real-time event tagging and stereo triggering, enabling coincidence detection across multiple telescopes. Even more stringent timing is necessary for intensity interferometry in the visible band, where fluctuations of light intensity from a source are sampled at multiple telescopes and correlated to reveal a source-size-dependent correlation peak, providing unique insights into the structure of astronomical objects. We explore the integration of White Rabbit (WR)-a fiber-based Ethernet timing technology developed at CERN, into the clock distribution network for intensity interferometry on optical telescopes separated by kilometer-scale baselines. Built on top of the Precision Time Protocol (PTP), WR offers sub-nanosecond synchronization with deterministic latency. We demonstrate that a WR network can provide sufficient synchronization accuracy over kilometer-scale distances while efficiently enabling time sharing between optical and IACT observatories for joint intensity interferometry observations. We investigate the effects of fiber temperature fluctuations and chromatic dispersion, which can impact long-distance timing stability during observations. Through lab experimental validation, we achieve 5 ps RMS time synchronization over a 5 km fiber link, and about 8 ps RMS for 10 km and 50 km links, demonstrating the feasibility of WR for high-precision astronomical applications like intensity interferometry.