I have the same screen where a replay can be saved, but the time doesn't seem to be saved. At the end of the stage there is a display where it shows the "Stage Best", "Personal Best", and "Current Time". Don't think I have the wording quite right, but there are 3 times displayed. I would have guessed that it would save my personal best so that if I come back the next day to run the stage again it would still be there for comparison...