How long do hot X-ray sources last? Some X-ray sources in nebulae are caused by hot materials in the millions of degrees that formed in supernovae. What's the cooling rate of this stuff? How long does it stay this hot?

If we would take our solar system for example, the time from the formation of the nebula (by a supernova?) until it was cool enough to form solid planet was only a few million years. That's very quick, for instance relative to the age of the Earth. How ephemeral are these X-ray sources? I would expect them to not last as long as IR sources.

Is there an order of magnitude for the lifetime of the emission? Does it take millions of years? Tens of millions? Tens of thousands of years?

A comment suggests it doesn't really cool down, but diffuse to the interstellar medium. How long does it take for it to diffuse until there isn't a clear X-ray source?
This is not an answer, but it is too long to put in the comment.

There are formulas for cooling timescales of different radiative processes. Cannot recall right now about sources for reading.

One thing about nebulae. Some of them might have compact objects such as pulsars or black holes feeding energy from the centers. In those cases, you have to consider the energy input from the central sources and radiative cooling.

Another thing about X-ray emission processes. There are thermal X-ray emission (i.e., bremmstrahlung) and nonthermal ones (i.e., synchrotron and inverse Compton; IC). Each process dominates in different environment. Regarding to SN nebulae, nonthermal processes are the dominant ones, the electrons involving in the processes are relativistic and thermal process requires dense electron medium (for collision) to maintain the thermal equilibrium. IC normally dominates at early ages of nebulae due to the compactness of the system, while synchrotron self-absorption is also strong at these ages. At older ages, synchrotron dominates, while the IC can interact with the background photons (i.e., cosmic background radiation) and results in X/gamma-ray photons as well.

Last about IR emission. Since IR emission is mostly from thermal process while X-rays are from nonthermal processes, X-rays tend to last longer.