In a walking distance from the Golden Pavilion you can find the Ryoan-ji Temple.
In this site you can find one of Japans most famous rock garden. In the rock garden are placed 15 rocks with small pads of moss... a very beautiful layout. It says ( I did not test it) that from any angle the garden is viewed only 14 of the 15 rocks are visible at any given time. The best time to visit the garden is in the morning as there are less people.
On the site there are also a big park with a lake you can walk around.