Closest pool is Cook and Phillip; literally 500m diagonally across the park.
It's not really a relaxing place though - lots of office workers doing laps and quite warm inside with the pool heating.
Your hotel is right on the bus route for Bondi, Bronte, Clovelly, Coogee, Maroubra, Malabar, Little Bay and La Perouse beaches... travel time from 25mins-50mins depending on route and traffic. Great way to cool off at the end of the day (best time to go down IMO is between 4 and 6pm)
I agree with aussieboy and suggest you visit the beach to cool off: head there in the later afternoon, go for a swim, have some dinner and come back. You'll also sleep like a log.
If you really want to swim in the city, the Boy Charlton pool is a great place for a swim, even if it is a little bit of a walk away.
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