To keep your rental from becoming money pit (whether you live there or are the landlord), there are a few things you can do. These are both ways to reduce heating costs and also the means of preventing costly damage caused by cold.
For instance, one major hassle which occurs in cold climates is that outside pipes freeze. As we all know, water expands when it freezes. Unfortunately, pipes do not. If they are forced to try, they tend to crack and cause internal water damage. Turning off and insulating outside pipes is a winterizing technique. Some people keep taps dripping to stop ice from backing up. This is also a fun way to make stalagmites.
Another home improvement idea in the same vein is cleaning gutters so that water flows without freezing. Again, expanding ice wrecks things (like gutters) which were not made to flex. Ice also blocks the path of water causing it to back up. Removing leaves, moss, and twigs in October is much easier than trying to get rid of gutter ice in December.
Now that you have turned the air conditioning off, it is time to turn the furnace on once more. Heating costs can mount up, but being cold is uncomfortable. Is there a compromise between freezing and going into debt? The compromise could be lighting a fire if you are lucky enough to have a functioning fireplace. As long as the chimney is clean and you remember to open the vent, lighting a fire is a cozy way to stay warm. Make sure you have enough logs to get you through the winter. Keep them dry (wet wood either fails to burn or smokes), and set apart very small pieces including bark for getting the fire going.
Change the filters in your air vents, the ones that catch crayons when they fall down the slats. They also catch dust and grime, helping you to maintain a healthy environment for lungs of all ages. If those filters are clean, heat is able to move through a home efficiently, reducing heating costs still further. If a room is not in use, shut the vent to keep air flowing where it is needed.
Some clever consumers even set their fans to blow hot air from its source to where it is needed most. If you have a table or ceiling fan, give this a try. At the very least, keep extra blankets in bedrooms and even on the sofa for cold nights in front of the TV.