We introduce a method to numerically compute the entanglement entropy of a one dimensional acoustic black hole. It is shown that the entanglement entropy of sufficiently large subregions scales linearly with size and thus shows a volume law instead of an area law. The origin of this scaling can be traced back to the non-separable long-distance correlations due to the production of phonon pairs at the horizon. The system is shown to be locally thermal, such that the part of the entanglement entropy scaling with volume is well approximated by the thermal entropy of the outgoing Hawking radiation.