The first thing you need to do is make sure that the battery is charging properly and is holding the charge. The battery itself may be weak and need to be replaced. Based on your question, I am assuming that you only have 1 twelve volt house battery on your RV.
You need to verify that the battery that you are using is a Deep Cycle RV/Marine Battery. If it is not a deep cycle battery then that is why it is running low when running the furnace. It is important that you use Deep Cycle Batteries in your RV. Please take a look at the answer I gave to the question below to find out the difference between normal batteries and Deep Cycle Batteries.
If you really want to ensure that you have sufficient 12 volt power in your RV your best bet is to use 6 volt Deep Cycle Batteries hooked up in series to power your RV's 12 volt system. This system will normally power your RV for twice as long as one 12 volt Deep Cycle Battery.
To understand why 2 batteries are better than one and to also learn how to setup the batteries read the answers I gave to the two questions below.
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