To start:
Invoke-Command -ComputerName "$REMOTE_SERVER" -ScriptBlock { Import-Module WebAdministration; Start-WebAppPool -Name "$APP_POOL_NAME" }

And to stop:
Invoke-Command -ComputerName "$REMOTE_SERVER" -ScriptBlock { Import-Module WebAdministration; Stop-WebAppPool -Name "$APP_POOL_NAME" }

Depending on what you need to do you might need to add an explicit wait time inside the scriptblock.

To start, sometimes you need to add an explicit wait so that the app pool responds to control messages:

Invoke-Command -ComputerName "$REMOTE_SERVER" -ScriptBlock { Import-Module WebAdministration; Start-Sleep -s 10; Start-WebAppPool -Name "$APP_POOL_NAME" } 

And to stop:

Invoke-Command -ComputerName "$REMOTE_SERVER" -ScriptBlock { Import-Module WebAdministration; Stop-WebAppPool -Name "$APP_POOL_NAME" }