beyond86 You need to look at the IEEE, NEMA, and IEC definitions of "cold" vs "hot" starts and what - exactly - that entails.
COLD START: ALL PARTS OF MACHINE ARE AT AMBIENT TEMPERATURE
HOT START : WHENEVER ANY PART OF MACHINE IS NOT AT AMBIENT TEMPERATURE
When a machine attempts to start the FIRST time, it is effectively at the same temperature as the surrounding medium (usually air). This is considered a "cold" start. During the start attempt, machine windings and other components will generate heat or become warmer due to simple proximity to a heat generator. The design of the machine (in terms of thermal mass and torque generation), coupled with the load characteristic (inertia, torque required to accelerate, acceleration time), determine how much heat is generated by a given attempt.
The "default" sequence is 2 cold, 1 hot. This means one attempt from ambient, plus one immediately afterward OR one attempt after the machine has reached a steady-state operating temperature below the maximum allowable temperature rise. Some designs use a 3 cold / 2 hot sequence ... which effectively means components don't get as hot during a given acceleration attempt.
The "start sequence" instruction from the manufacturer usually includes some sort of discussion regarding how long a dwell time is necessary after a "hot" start the next attempt can be made. The duration has to do with whether the machine's rotor is spinning at or near running speed, thus moving air within the machine enclosure. For most machines with a rotor turning at or near operating speed, the dwell time is 20-30 minutes. For machines where the rotor is at standstill (i.e., the machine stalled and never got turning in the first place), the dwell time is typically 45-60 minutes.
Further to your initial discussion, I also see you want to perhaps start the motor in a (relative) hurry after a shutdown, presumably to limit difficulties with the driven load (compressor) such as condensation. In this case, you'd also need to know how long the motor takes to spin down to zero speed when connected to the load (should be on the order of seconds, if not tens of seconds) AND whether the machine can be restarted "on the fly". There is a very short period of time where you can reclose the breaker and "catch" the motor ... for something in this power/speed range, that is likely to be under six seconds. If you cannot reclose successfully in less than that period, you will have to wait until the unit comes to a complete stop. If the machine has reached steady state operating temperature before the event, you would have one less than the listed number of "hot" start attempts to try without waiting for the defined dwell time.
Lastly - there is often one other line to the starting discussion. That is the MAX Number of start attempts allowed in a 24-hour period. Typically, this is SIX. In case you're interested, that comes from the wear and tear associated with the start itself. Most machines are designed to handle one start per day, five days a week, fifty weeks a year, for twenty years (i.e., 5000 starts). If you actually tried to operate at 6/day, your machine life expectancy would be about 27 months.
Converting energy to motion for more than half a century