I am not sure the Hawthorn area, but i know some pools which are around the city (and I used to visit before), not sure if it is ok with you.... the info is as follows:
1. Melbourne City Bath (more than just a pool) (indoor). Cnr Swanston Street and Franklin
Street.
2. Melbourne Sports and Aquatic Centre. (indoor) Aughtie Drive Albert Park, Ref: Melway 2K, D7
hope it will be any help of you!