Is there a law of physics that would prevent a material from keeping a constant average temperature of it's entire environment over it's entire surface?

I know it hasn't been done yet, or possibly it would be a feature of superfluids.

And just to be clear what I'm talking about. A material that when you put an ice cube on one end and a blow torch on the other, would maintain a temperature halfway between the two extremes, even on the ends directly in contact with the flame or ice. Think copper with a near 0 time delay for heat spread.

It would definitely be useful, but because the ice cube has less cooling power than the blow torch has heating power, I think the spread would be higher than halfway, unless you had a self regulating system, which would defeat the purpose.

I've always wanted a porpoise.

Granted the ice cube wouldn't last long and the whole thing would reach near blow torch temperatures shortly after.

After looking into it a bit more I'm not sure it can be done, or we would even have the technology to try. It would require a material that acted like a single crystal over it's entire length and translated energy equally to every atom at once. I haven't seen anything that can transfer kinetic energy that efficiently. Diamond is one of the closest as far as atomic bonds goes, but even it works as an insulator.

Thermal Superconductor

Superconductive Power transistor would be possible and be a very valuable component. Playing around with the ceramic superconductive materials and treating them with P type and N type elements used in regular PNP NPN transistors could produce something functional.

Applications could be used in high power burst transmission of radio signals and telemetry signals. What that means is a 10 minute message would be compressed into a 1 second burst transmission from a compact devise that would have long range because of the amount of power the superconductive transistor could handle without blowing from heat produced from the amount of power passing through it.

Granted, electrical superconductors have a billion and one uses where they would help. I was looking more at thermal/kinetic energy exchange.

I'm not even sure it could be done really since even solid matter isn't actually solid. The atoms are simply locked into position by their electrons. But that doesn't give them the ability to transfer thermal energy directly without a delay.

Superconductive material in terms of momentum passing through it is interesting.
Consider a metal cube (1 cubic meter) with a steal ball touching one side. Grabbing a sledge hammer and banging the other side of the cube would transfer Kinetic Energy in terms of `conservation of momentum' to the steel ball. However the speed of the transfer of momentum from atom to atom within the cubic meter cube depends on something like the `speed of sound' equivalent to a momentum wave passing through a given type of metal.

Freezing the material with liquid nitrogen wouldn't help and would probably cause the material to shatter like the `Judgment Day Terminator'

The thermal transfer of Energy along the lines of heat as we understand couldn't be done superconductively because heat is random incoherent motion of atoms. Coherent transfer of heat would be in metal rods of an X-ray laser.

How fast does/can an electrical wave transfer from atom to atom in a superconductive liquid nitrogen cooled superconductor? Good question because if it can be made to be the speed of light as i would expect could be possible then you could have a piezo electric superconductor connected to a superconductive wire with a superconductive transducer at the other end. Kinetic Energy would be converted to a speed of light electromagnetic wave at the piezo side, travel ling through the superconductive wire and then being converted into Kinetic Energy again by the superconductive transducer.

Heat in incoherent random bouncing around of atoms. Your quest of a kinetic energy superconductor could lead to building a bell that would resonates coherently and superconductively at the speed of light. The atoms wouldn't be moving but the standing wave energy shells of the atoms would vibrate from side to side around each nucleus of each atom. Imagine a metal rod that resonated like a bell from end to end coherently like a laser but without emitting photons

Thermal superconductivity is possible because `phase state teleportation' between `quantum entangled materials at distances' is possible and has been demonstrated in laboratories along with `quantum computers' utilising the effects.