Originally Posted by TA
And are they relying on the other engine to restart in case anything goes wrong in the air?
Probably not in this case.
I don't know the specifics on the A319 series, but there are usually several options for engine restart or aux. power in the air that are not the APU. For one, the airflow through the engines will be enough to turn the turbine blades, though I'm not sure if that alone is enough for a restart.
Many aircraft also have a ram-air turbine, or RAT, that can be deployed in flight. This is essentially a small "windmill" that can be deployed and uses the airflow across the moving plane to generate enough power for emergency systems.