In a way, yes. the amp does not know the power handling or voice coil size of the speaker, simply "sees" the impedance and has a designed output potential, based on that impedance. heck, amps dont even know how many speakers are connected, again, just the load placed on the chnl(s). We can power 1 or a 100 speakers, as long as the impedance is within the amp safe limit. Power would suck to 100 speaker per chnl, but the amp doesnt care :p Luckily, the gain and volume dials can be used to regulate the amp's output, up to its potential.
If this amp has even half its stated RMS, it should NEVER go into thermal shut down driving a pair of 4 ohm 8" coaxials. Even being class-AB and in death valley, the gains should be pretty conservative. If it is a thermal event, it will take more then just lowering the volume to get the amp back on line. The amps usually need a few minute or more to cool. Not ruling out, id just place heat down on my list.