Place first 3 ingredients on a double layer of cheesecloth. Gather edges of cheesecloth together; tie securely.
Combine cheesecloth bag, cider, and next 5 ingredients (cider through salt) in a large stockpot; bring to a boil. Reduce heat, and simmer 20 minutes. Discard cheesecloth bag. Serve with rum, if desired.
This makes a large, delicious pot of cider. However, I'd recommend buying cheesecloth instead of trying to use a tea bag. My tea bag broke midway through simmering and I had to strain the cider into a few different pots (since it makes such a large quantity). Regardless of the trouble, though, the end result was worth it. Especially with rum added, it makes a hot and cheery winter drink.