The radiofrequency resonant four-quadrant mechanical structure of a 4-Vane Radio Frequency Quadrupole (RFQ) has a high quality factor and narrow bandwidth, resulting in high sensitivity to frequency detuning caused by thermal expansion under high power operation, so minimization of frequency errors and affording dynamic tuning are very important design issues. Here we describe an optimization approach to figure out a suitable cooling design for a 4-vane RFQ in steady state. Besides, we investigate how the accelerator responds in transient thermal analysis that could help to guide commissioning and reduce frequency detuning. Multiphysics analysis utilized with CST and ANSYS for a recently developed 176 MHz 4-vane RFQ is taken as an example. This RFQ will dissipate 211 kW when reaching an inter-vane voltage of 80 kV, which is required for an acceleration of an 80-mA proton from 65 keV to 2.5 MeV in 5.3 meters.