Adjust oven rack to middle position and heat oven to 350°F/180°C. Grease well a 12-cup (10-inch) bundt pan and set aside.
To make the cake: In a medium bowl, whisk together flour, cornstarch, baking powder, and salt. Set aside.
In a standing mixer fitted with the paddle attachment, beat together butter, oil, and sugar for 5 minutes until light and fluffy. Occasionally scrape down the sides and bottom of the bowl. Add eggs, one at a time, beating well after each addition. Add heavy cream (or milk), rum, and vanilla, and beat until combined. On low speed, beat in the flour mixture and mix just until combined.
While the cake is in the oven, make the syrup: In a small saucepan, combine butter, water, and sugar. Cook on medium-low heat until sugar dissolves and syrup forms. Turn off heat and mix in rum. Set aside until the cake is ready.