I am not aware that SSAS stores the last started value anywhere. If you are ok with using the servers event log, a PowerShell scripts can get it for you:

(Get-EventLog -LogName "System" -ComputerName <Your Server Name> -Source "Service Control Manager" -EntryType "Information" -Message "The SQL Server Analysis Services (MSSQLSERVER) service entered the running state." -Newest 1).TimeGenerated;