Check Your Network Connection

Ensure that your PS5 is connected to a stable and reliable internet connection. Try restarting your router and reconnecting your console to the network.

Adjust Power Settings

Go to Settings > System > Power Saving > Set Time Until PS5 Enters Rest Mode. Make sure it's set to a reasonable time, and not too short.

Update System Software

Make sure your PS5 is running the latest system software. Go to Settings > System > System Software > System Software Update and Settings.

Enable Background Downloads

Go to Settings > Saved Data and Game/App Settings > Automatic Downloads and ensure "Enable Automatic Downloads" and "Application Update Files" are checked.

Conclusion

By following these troubleshooting steps, you should be able to resolve the issue of your PS5 not downloading in rest mode. If the problem persists, it's advisable to reach out to Sony's support for further assistance.