Why must there be a long period (measured in hours) between shutdown and startup, before using the manual engine start valve? Nine hours seems excessively long.
The FCOM page reports MSN 7172 which corresponds to an A320neo with PW1127G engines. Why is this important? Because this engine family is prone to bowing, so much that it requires a dedicated cooling procedure on start performed by FADEC:
However, the ENGINE START VALVE MANUAL OPERATION procedure requires you to perform the MAN START of the engine, which disables many safety feature of the FADEC and does not permit the FADEC to close the start valve in case of the detection of an abnormal situation (this is actually the reason why you want a MAN START). So, I suspect it also disables bowed rotor protections, and this is why that procedure requires the engine to be completely cooled down.
If you are wondering how is possible that 9 hours are necessary for the engine to cool down, consider that such engine requires about 2 minutes of forced cooling even after 1 hour since shutdown (source). The most critical point for is around 2 hours since shutdown (source). In some forum you can find pilots reporting ~6 hours the needed time for residual EGT to go to ambient temperature for LEAP engine, which is even less problematic for thermal characteristics compared to PW.
So it's reasonable to think that the engine core with its massive thermal capacity may require many hours to reach ambient temperature, and the need of MAN START probably disable safety features that would avoid bowing, which does not allow you to start the engine if it's still hot.