The 4.0 APK is a temporary fix. Because Google constantly updates its authentication API, even the 4.0 version may eventually be phased out. If you want long-term usability on legacy hardware, consider:

Custom ROMs: If you have installed a minimalist custom ROM that didn't include Google Apps (GApps), you may need to install the manager manually. How to Download and Install Google Account Manager 4.0

To understand why someone would specifically look for version 4.0, it is essential to understand the function of the app itself. Google Account Manager is a system application that comes pre-installed on almost all Android devices.