Nothing beyond the £500m cost was contributed by ScotGov, despite the project ultimately costing £1bn+.
If you actually look at council funding, Edinburgh gets the lowest per head in Scotland: https://www.gov.scot/publications/scottish-local-government-finance-statistics-slgfs-2020-21/pages/3/
The average per head is £2136. Edinburgh gets £1771, while GCC gets £2290.