In a large pot add the cooked rice, water, cinnamon sticks, coconut milk and bring to a boil. Simmer on low for about 20 minutes stirring occasionally until the rice has broken up and thickens, add the sugar and simmer for another 10 minutes or until desired consistency. Add more water if it is too thick. The texture will thicken when it is cooled and it will also be a little less sweet once it is refrigerated.
Allow to cool, and then refrigerate until ready to serve.