According to this, The Royal Hotel is the most common* name for an Australian pub, which, from a number of road trips I've taken over the years, rings true, especially in NSW**. There's one just a few blocks from where I live, and such is its ubiquity it was used as both the pub name and the title of a recent Australian film. The map below shows all --- Royal Hotels in Australia***. The closest are only - apart (The Royal in Darlington and The Royal in Leichhardt, NSW) and the furthest, on Thursday Island in the Torres Strait, is - away from its nearest, which is in Herberton, south west of of Cairns (the average distance from one Royal to its nearest neighbour is only -).
Drag the on the map below to see how far a location is from the nearest Royal (the furthest you can be from a Royal on the Australian mainland is just over 2,000km). The shaded areas show the parts of the country where you are within 100km of a Royal (- of Australia).
A Royal Hotel The ones I've been to Drag (centre marker)
Made with Leaflet, Turf.js, Mapbox, & Geoapify. Some addresses were checked with Google Maps. Map copyright OpenStreetMap contributors.
List of all locations with a Royal Hotel (ones I've been to in bold): -.
* Second most common was The Commercial
** Number of pubs named The Royal by state were:
*** For this I only used pubs just called the Royal Hotel, so pubs such as the The Royal George Hotel, The Royal Exchange Hotel, The Royal Albert Hotel etc were excluded. Pubs with the town or suburb in the name such as The Royal Darlington, were included.