Preheat oven to 325°F. Grease a 9-inch springform pan with butter or cooking spray.
Combine graham cracker crumbs, carrot, melted butter, and sugar in a bowl. Press mixture onto the bottom of the prepared pan using a flat bottomed glass. Bake for 7-10 minutes, then let cool completely.
In a large mixing bowl, beat the cream cheese and sugar with an electric mixer on medium speed until smooth and creamy. Add the coconut milk, eggs, and chia seeds, and mix until well combined. Pour the mixture over the cooled crust.
Bake for 50 minutes or until the center is almost set. Cool completely at room temperature. Then chill in the fridge for at least 4 hours or until the cheesecake is firm.
Meanwhile, make the topping. In a medium saucepan, combine crushed pineapple, rum, sugar, and cornstarch. Cook over medium-high heat, whisking constantly, until the mixture thickens and turns glossy. Remove from heat and let cool.
Spread the cooled pineapple topping over the cheesecake and serve.