  Topological phases have prevailed across diverse disciplines, spanning
electronics, photonics, and acoustics. Hitherto, the understanding of these
phases has centred on energy (frequency) bandstructures, showcasing topological
boundary states at spatial interfaces. Recent strides have uncovered a unique
category of bandstructures characterized by gaps in momentum, referred to as
momentum bandgaps or k gaps, notably driven by breakthroughs in photonic time
crystals. This discovery hints at abundant topological phases defined within
momentum bands, alongside a wealth of topological boundary states in the time
domain. Here, we report the first experimental observation of k-gap topology in
a large-scale optical temporal synthetic lattice, manifesting as temporal
topological boundary states. These boundary states are uniquely situated at
temporal interfaces between two subsystems with distinct k-gap topology.
Counterintuitively, despite the exponential amplification of k-gap modes within
both subsystems, these topological boundary states exhibit decay in both
temporal directions. Our findings mark a significant pathway for delving into k
gaps, temporal topological states, and time-varying physics.
