There is no GUI available in the SharePoint Central Administration website to delete a SharePoint 2010 application pool.

Deleting the application pool from the Internet Information Services (IIS) manager is also not helping as you will still get the error when trying to create a new application pool with the same name.

The only way is to delete the application pool entry from SharePoint is via PowerShell command.

Here are the steps:

1. Launch the SharePoint 2010 Management Shell with administrator privileges
   (Start -> All Programs -> Microsoft SharePoint 2010 Products -> Right click on SharePoint 2010 Management Shell -> Run as administrator)
2. Run 'Get-SPServiceApplicationPool'
3. Run 'Get-SPServiceApplicationPool -Identity <Name of the application pool>'
4. Run ‘Remove-SPServiceApplicationPool’
5. Input the Identity <name of the application pool>
6. Input 'Y' to delete the application pool