Combine the sugar and hot water in a heavy saucepan, and cook over high heat until the sugar has dissolved, about 1 minute.  Continue to cook, swirling the pan occasionally but never stirring the sugar mixture, until the mixture thickens and turns a deep amber color, 5 to 8 minutes.  Remove the pan from the heat and immediately add the heavy cream (be careful--the mixture will splatter).  Return the pan to the heat and reduce the heat to medium.  Cook until the sauce is thick and creamy, stirring occasionally to help incorporate the cream, about 5 minutes.

Remove the pan from the heat and stir in the rum, vanilla, and butter.  Serve warm.  (If making the sauce in advance, rewarm gently before serving.) It will keep for several weeks refrigerated in an airtight container.