Late autumn, winter and early spring (April to September) are definitely the best times to climb in the Sydney region. The temperature almost never drops below zero degrees Celsius in the Sydney basin, and the weather is usually pretty stable during this time. It can get bitter at shady crags that catch the wind (eg. Bangor), but generally it's always possible to climb during the cooler months. Friction on sandstone is also at its best when it's cold.
Climbing in summer is a real battle against the grease. It tends to be unbearably hot (35+ degrees Celsius is pretty common) and wet as well, with stinking hot humid days followed by seemingly endless days of torrential downpour that are barely any cooler. The shady south and east facing crags are the pick during the hot months, with Narrabeen Slabs, the Junkyard Cave and the Cathedral being good options. The sea cliffs are also a good option, since they are usually in shade by mid afternoon and swimming is usually possible. The boulders on Palm Beach are probably the pick of these areas.
Another disincentive for climbing during summer is the sea slime, which coats most of the sea cliffs and crags near the ocean. This can also happen at other times of the year, particularly if the prevailing weather is easterly.
Foreign visitors should be particularly careful not to underestimate the power of the sun. Good sun protection (sunscreen, hats etc.) is an absolute must during summer. Now that you're in the southern hemisphere, note that north facing crags get most of the sun (great for winter, crap for summer), and likewise south facing crags get little sun and are slower to dry out.
The sandstone found in Sydney is quite porous. Many of the crags seep during and particularly after rain. The rock also has a tendency to soften when wet, so take care if you're cranking on small holds after rain. Damp rock will break and your protection will fail.
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