We discussed this possible application in another thread. Delivering a constant voltage irrespective of temperature may be admirable for a voltage reference, but for biasing up a power amp output stage the commonly used Vbe multiplier (one transistor and two resistors) will exhibit a similar voltage shift with changing temperature as the output power devices, desirable to prevent thermal runaway. These Vbe multiplier bias transistors are routinely thermally bonded to the same heat sink as the power devices so they will track.
I don't doubt somebody stuck a TL431 inside a power amp output stage, I've seen worse things. Driving the output stage with a constant bias voltage, means the class A current will increase as the power devices heat up and their Vbe gets smaller, increasing the class A current even more and increasing the heat dissipation, dropping the Vbe even more... you can see where this is going. This can realize thermal equilibrium without releasing smoke "if" A) the emitter degeneration resistors are large enough to limit the class A current to some tolerable power dissipation, and/or B) the heat sink moves enough heat out to prevent too much temperature rise and Vbe drop. I recall seeing equations to quantify this (decades ago) but some of the equation terms are hard to put a finger on (like thermal resistance to ambient).
The TL431 IIRC is already borderline too much voltage for typical bias applications but perhaps the low voltage version (1.25V?) could use a temperature dependent resistor in the voltage setting feedback string to move the bias voltage in the correct direction with temperature change.
When I was product manager over all power amps for Peavey I inherited a value 2kW class A/B amp. Since this design scrimped on power devices to reduce cost it was borderline stable thermally and the service repair record reflected that. I added a pair of metal TO-3 devices per channel to lower the thermal resistance (junction to sink, and ultimately to ambient) to stabilize the amp. Adding several dollars to the BOM of a value amplifier was not embraced by Peavey management, but it was the right thing to do. These days a 2kW class A/B amp is a relic of times past.
At Peavey a widely used approach for the output stage bias string involved using a specific dual diode with a tightly specified temperature/voltage characteristic that was similar to the metal TO-3s used in large numbers. The dual diode was routinely inserted through a hole drilled/punched in the aluminum heatsink L-bar mounted on top of the PCB, and soldered in place.
PS: I did a great deal of bench work last century designing a bias circuit that targeted regulating the actual class A current ignoring temperature. I had a working circuit but the engineering manager I reported to at the time lacked the cohones to let me use it in a production SKU . I still regret his bad judgement. (Another reason I promised myself to never be an employee again if i can possibly help it). I built my prototype bias circuit into a cheap guitar amp using plastic darlington power devices that was tragically under biased when cold... the stock amp would literally sound like crap until it warmed up and realized decent class a bias. Not very good for POS demonstrations in the dealers showrooms on cold winter days. Sorry if I am repeating myself, even this trick bias circuit is documented elsewhere on this site.