This is the unfortunate impact of cold weather on a hybrid vehicle. The first priority of the car is to keep the engine up to temp so that it can run efficiently, which is why it starts out running. Its second priority is to give you heat. Since it would be impractical to us the battery to generate heat, the source is the engine. That means the engine has to start up and burn some gas to generate heat to keep both you and itself happy. This puts fuel economy in third place.

The explanation for the engine running after the engine is up to temp is the state of the battery. NiMH batteries don't like to give or take a charge when they are below about freezing, so the system won't use it. Instead it keeps the engine running to drive the vehicle while it allows small currents to pass through the battery, which gets it warm along with cabin heat. This helps to give the battery a long life.