In short, there is no approval mechanism for Time Entries and you cannot add fields to the Time Entry tab (i.e. to the TE records in the database).
Basically, the Time Entries fields give you a suite of numbers tied to an item, but without states and ways to customize them.
This means that if you want to have an approval process, you are going to have to develop it yourself.
My first idea would be to have an approval State, or a Pick List, or Boolean field, etc. on the item to signal the TE approval status. When the item is approved, I would have a computed field or one calculated by trigger that would hold the total of Time Entries. So for example, my item would have an "Approved TE Total" and "Pending TE Total" that I could use in reports (note: mutually exclusive). You could even add a "Rejected TE Total".
This would be for the total entries for an item.
I cannot think of a simple way to have approval of individual entries or by period (e.g. weekly) if that's what you are asked to do.
