Whisk together rum and honey. Divide between the grapefruit halves and pour all over the top. Sprinkle with brown sugar and cinnamon. Let it sit at room temperature for 30 minutes so the flavor of rum, honey, brown sugar and cinnamon can permeate the meat of the grapefruit.
Bake at 350 degrees for 30 minutes.
Broil for about one minute or until the tops start to brown.
Serve warm. Save a little honey to drizzle on the top of the grapefruit if needed.