Always On - keep secondary database readable when connection to primary node failed
Is there a way to change this behavior and keep secondary databases always online? 


Yes.  Use SQL 2014 or later :)


  Beginning in SQL Server 2014 (12.x), readable secondary replicas can
  remain online even when the primary replica is offline due to user
  action or a failure. However, read-only routing does not work in this
  situation because the availability group listener is offline as well.
  Clients must connect directly to the read-only secondary replicas for
  read-only workloads.


Active Secondaries: Readable Secondary Replicas (Always On Availability Groups)