All ingredients should be room temperature, otherwise you will have lumps. Heat your oven to 200. Get out a deep cake pan and butter it. You'll also need a large roasting pan to put the cheesecake pan in.
Beat the cream cheese until it's smooth and fluffy, scraping the bowl and beaters to be sure. Then add the sugar, vanilla, cognac, and rum and beat until incorporated. Now add the eggs, one at a time, beating until each is incorporated.
Pour this mixture into the cheesecake pan. Put the cheesecake pan into the roasting pan and carefully pour hot water into the roasting pan until you have a couple inches of water. Then carefully put all of this into the oven and set the timer for about 4 hours.
After 4 hours, check your cheesecake. Does it look dry on top? Is it not too jiggly? Is its temp 180? Does the thermometer come out clean? If you can answer yes to all these questions, your cheesecake is probably ready. If not, let it go another hour or two, checking every once in a while.
Once you feel good about the doneness of the cheesecake, take it out of the roasting pan and let it sit at room temperature for a while until it is completely cool. Then unmold it onto a plate and refrigerate it for a few hours or overnight.