Controlling a PSC motor with a triac thermostat
It is difficult to confidently determine how the Aube TH401 works. The 15 second and 5 minute cycle mode selection terminology seem to indicate that both modes use zero crossing or integral cycle control as described in Transistor's answer to this question. That seems the most likely case, but the "heating intensity" indicator confuses things a bit. It looks to me like the "Std" setting results in a 15 second minimum on and off times in response to the thermostat. In addition there are 5, 10, 15, 20, 25 and 30 minute settings. That provide longer on and off times. The manual does not have a very good explanation.

The product information indicated the device is not rated for inductive loads. That seems to contradict the information about a "fan" setting. The device may be ok for an electric heater with a small fan. A load like that would be predominantly resistive. The product indicates it is not for driving a relay. There is also a minimum current that is higher than the current required for most relays.

The bottom line is that the device will probably not control the speed of a PSC pump. It does not even seem to be recommended for turning a PSC pump on and off. If the device is ok turning a pump on and off, the pump motor would probably be ok with 5 minute or longer cycles, but it would probably last longer if the cycles were a little longer than 5 minutes. If the device would actually modulate the speed by phase angle control, that would probably be ok (but no guarantee) if the lowest speed was not slower than the lowest switch speed. The speed switch would be set to the highest speed when using a phase angle control unit.