You would use a cookie to store the information in to client side.
You would need to store a reference to when the stopwatch / timer was started, a reference to the time when the page was closed and a reference to when the timer is to be considered lapsed and or ended.
So each load of the page would result in the cookie being loaded, the existing data deleted (as in delete the cookie) then on page close set a new cookie with the above information.
All you need to do is google setting and deleting and reading cookies, how to run a function on web page close and your timer / stop watch.
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