Dell has changed the name of its acquired AppAssure to Dell Data Protection Rapid Recovery. AppAssure (latest version is 5.4.3) is no longer being maintained, so upgrading to Rapid Recovery (latest version 6.0.2 at the time of writing) is a requirement for continued support. Fortunately, performing the upgrading is not really difficult, and can be done without too much downtime for your backup.
Additional information, including a compatibility and upgrade matrix can be found here:
At the time of writing, the compatibility matrix shows that Agent version 5.4.3 can connect to Core version 6.0.2. As such, if you can upgrade the Core before upgrading any Agents, backup will continue to work after the upgrade of the Core, if the Agent's version is the latest (version 5.4.3). However, it is strongly recommended to upgrade the Agent as soon as possible. It is not possible to upgrade the Agent prior to upgrading the Core.
If you use replication between different Core servers, the target Core must run the same or a higher version than the source Core. For a seamless upgrade, upgrade the target Core before upgrading the source Core, although it is highly recommended to upgrade all Cores as soon as possible.
Perform these steps in order to avoid loss of connectivity of your Agents to the Core(s).
Review How to Upgrade AppAssure 5.4 to Rapid Recovery 6.X (181997) for detailed steps.