What happens to the backup time when batteries are connected in series?
Putting two batteries in series doubles the power available over that of one battery. 

Whether your load is capable of using that power efficiently is another matter. 

In the case of an inverter designed to run from a specific voltage, you can probably assume it will use that input power efficiently. For the same load, for equal efficiencies, a 24v input inverter will draw half the current from the batteries that a 12v input one would.