Preheat the oven to 200 degrees F. Place pineapple on a silpat (a French nonstick baking mat) or parchment lined baking sheet. Sprinkle with 1/3 cup sugar and bake, checking every 30 minutes, until juices become syrupy, about 2 hours. Set aside.
In a heavy-bottomed saucepan bring 4 cups cream and rum to a gentle boil. Add pineapple, and remove from heat. In the bowl of an electric mixer fitted with the whisk attachment, whisk yolks and remaining 1 cup sugar until pale and thickened, about 2 minutes. Place cream mixture over medium heat and bring just to a boil. Whisking constantly, add about 1/2 cup cream mixture to lightened yolks. Return to saucepan and continue to whisk until thickened, about 3 minutes. Set aside to cool.
In the bowl of an electric mixer fitted with the whisk attachment, beat the remaining 1 cup cream until soft peaks form. Gently fold into cooled yolk mixture. Transfer to an ice cream machine and freeze according to the manufacturer's instructions.