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This is what I get by using bench.py script attached to issue13451:
CURRENT VERSION (NO LOCK)
test_cancel : time=0.67457 : calls=1 : stdev=0.00000
test_empty : time=0.00025 : calls=1 : stdev=0.00000
test_enter : time=0.00302 : calls=1 : stdev=0.00000
test_queue : time=6.31787 : calls=1 : stdev=0.00000
test_run : time=0.00741 : calls=1 : stdev=0.00000
LOCK WITH DECORATOR (no synchronization)
test_cancel : time=0.70455 : calls=1 : stdev=0.00000
test_empty : time=0.00050 : calls=1 : stdev=0.00000
test_enter : time=0.00405 : calls=1 : stdev=0.00000
test_queue : time=6.23341 : calls=1 : stdev=0.00000
test_run : time=0.00776 : calls=1 : stdev=0.00000
LOCK WITHOUT DECORATOR (always synchronized)
test_cancel : time=0.69625 : calls=1 : stdev=0.00000
test_empty : time=0.00053 : calls=1 : stdev=0.00000
test_enter : time=0.00397 : calls=1 : stdev=0.00000
test_queue : time=6.36999 : calls=1 : stdev=0.00000
test_run : time=0.00783 : calls=1 : stdev=0.00000
Versions #2 and #3 have the same cost, so it's better to get rid of the explicit argument and always use the lock (version #3).
Differences between #1 and #3 suggest that introducing the lock obviously have a cost though.
It's not too high IMO, but I couldn't say whether it's acceptable or not.
Maybe it makes sense to provide it as a separate class (SynchronizedScheduler).