We explore the gapless topological phases of a $p$-wave superconductor, probing its rich topology and underlying quantum phenomena. The topology of the system is characterized by studying its entanglement properties. This study confirms the bulk-boundary correspondence in the entanglement spectrum, even without a full bulk gap. For contractible bipartitions, the entanglement entropy varies nonmonotonically with the chemical potential, displaying pronounced peaks at points where the bulk gap closes and reopens, signaling topological quantum phase transitions. This behavior remains robust in the thermodynamic limit. The entanglement entropy grows with system size for noncontractible bipartitions, obeying volume law even in the gapless phase. These findings reveal the subtle interplay between symmetry, entanglement, and topology in gapless systems, and emphasize the role of entanglement-based diagnostics in identifying unconventional topological phases beyond the gapped paradigm.

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• Entanglement entropy
• Entanglement measures
• Topological order
• Topological phase transition
• Topological superconductors