Since timesheet is not a studio object so you are not able to enable any type of audit. Anything you create to monitor the status of your timesheets would be custom via portlet. Below you can see some information that could help you with that.
How can the last edited time be obtained before a timesheet is posted?
1. The PRTIMESHEET.prmodtime will be updated when the timesheet is edited, saved, approved, or returned.
2. After the Post Timesheets job processes a timesheet
a. PRTIMESHEET.posted_time will be set
b. PRTIMESHEET.prmodtime will get set to the PRTIMESHEET.posted_time
3. If the goal is to obtain the latest
PRTIMESHEET.prmodtime when it approved, a customization will need to be considered.
PRTIMESHEET.PRMODBY is the resource who last performed an action on the timesheet.