I have a seemingly simple question about the relation between these three that for some reason doesn't make sense to me. If entropy is the disorder of a system, then a low entropy state is one of higher energy. As we know, mass is energy. From here we must say that the more mass something has, the lower its entropy because the mass can be converted to energy. Why then are black holes, the most massive things known, considered to be of such high entropy?
There are a few concepts to go through regarding entropy in thermodynamics - it's a tricky subject easy to conflate with probabilistic interpretations.
Temperature vs Heat
Temperature refers only to the particles' speeds. Increasing the temperature of an isolated system will increase the speeds of the particles, but does not of itself affect the entropy per se.
Contrast this with Heat, which most times is what is meant by talking about the internal energy of a system. The Heat of a system, is the total thermal energy - a warm box of gas has more heat than a cold one. But, Heat also relates to the quantity of particles - a large, warm box has more heat than a small box of the same temperature because there's more mass at that temperature. I don't know if it's right or not, but it helps me to think of Heat as 'thermal momentum' analogous to temperature being 'thermal speed'.
The difference is important, since the Work done by a thermodynamic system is proportional to a change in Heat content. That is, if you cannot construct a heat gradient, no work can be done.
Entropy vs Internal Energy
For something like Maxwell's Demon, you need to consider what happens to the heat content.
In a 'normal' situation the speeds of particles in a box of gas aren't going to be equal, but they'll approach a distribution that tends toward the average temperature. This means there is a set of particles at a lower-than-average speed and a set of particles at higher-than-average speed, but these particles are not separated so no work can be extracted - both sets of particles are occupying the same volume (the whole box of gas) so no significant heat gradient can be found to extract work from.
Maxwell's Demon, is a door / gatekeeper etc, that sifts particles such that these two sets end up apart from one another - say, the 'faster' particles on the left and the 'slower' particles on the right. The total energy has not changed (ignoring how the selection process works), the internal energy has not changed (particles are in different places but with their initial speeds), so what has changed?
The entropy - due to the re-arrangement, the internal energy (Heat) has been partitioned creating a gradient where there previously was none. This is despite the heat energy existing before in the same quantity. This means that a decrease in entropy, increased the amount of available energy in the form of heat.
Consider then, if the temperature distribution of the particles were larger - the same decrease in entropy would make available a larger proportion of the internal energy, as there would be a larger gradient possible.
Clearly, entropy relates to available internal energy. In this case the total internal energy is Heat, which is proportional to temperature.
Entropy vs Mass / Energy
Entropy on its own is not a measure of energy or mass, in fact mass itself is better related to Heat - changes in Heat correspond to changes in energy (and therefore to changes in mass).
Information vs Entropy
In Maxwell's Demon, this entropy isn't appearing from nowhere; we can't start with nothing and end up with something. In other words, what does the demon do to the system in a thermodynamic sense?
It adds 'information' to each particle it 'sorts' to either the left or the right. This same 'information' is then 'recovered' when the resulting liberated internal energy can be put to use.
In other words, the available internal energy (resulting from entropy / physical configuration) is in a very real sense describing the 'information' content as opposed to 'noise' or 'disorder'.
To put it very simply - the more information is encoded in a physical system, the less entropy it has, or if you prefer the term, less disorder.
Black Hole Entropy
It is worth revisiting the 2nd law in context of the previous conclusion - entropy tends to increase, would imply that information tends to decrease.
A black hole is a collapsed object - one crushed into a point by it's own gravity. As a result there is no way to represent physical information that has fallen into the hole, as should any fall in the matter / energy making it up is crushed beyond recognition.
Then, if no information can survive (or be used) inside a black hole, it suggests that it is a state of maximum disorder - a state of maximum entropy.
If decreases in entropy increase available energy, why do black holes have large entropy?
To answer your question directly - changes in entropy do not correspond to changes in total energy. Lowering entropy makes it possible for internal energy to become useful, but this does not create internal energy or change the total energy.
Conversely, increasing entropy will make it harder to extract internal energy within a system until a point where no internal energy can be utilized at all - such as inside a black hole. Per above, changing entropy will not change the total energy, so you cannot form a black hole by changing entropy; in this context entropy is a measure of a black hole or some object's available internal energy and nothing more.
Drop a glass on the floor -> Entropy increases. Drop another glass on the floor -> Entropy increases again.
From above we conclude that a big pile of broken glass contains more entropy than a small pile of broken glass. We also happen to know that a big pile of broken glass has larger mass than a small pile of broken glass.
Drop a glass on the floor -> Entropy increases. Hammer the pieces of glass to smaller pieces -> Entropy increases again.
From above we conclude that breaking a solid to very small pieces causes a large increase of entropy. We also happen to know that breaking a solid to very small pieces increases the internal energy of the (former) solid.
A quote from comments:
how is a higher temp and faster moving particles not an increase in available energy and therefore a decrease in entropy?
Maybe higher temperature means more available energy and more unavailable energy! It definitely sometimes means that.
We have devices that decrease entropy of heat energy, in other words they make heat energy more "available". Heat pumps are those devices. They pump heat to higher temperature, in other words they make heat less entropic.
We also have devices that produce heat energy and entropy, Stoves and radiators are those kind of devices.
So here's advice to OP: Do not make too general conclusions, as entropy can be removed from heat energy, which will cause a rise of temperature. But also more heat and more entropy can be added to heat energy, which will cause a rise of temperature.
Calling entropy "disorder" is somewhat misleading, it can also be described as the amount of area containing an amount of energy or even information. A black hole will contain a significant amount of quantum data in an incredibly small space, making them objects with high entropy. However this is a fixed amount of entropy and to continue the propagation of energy as is needed by the laws of thermodynamics, some of the particles containing this information will escape (look up hawking radiation).
Backing away from quantum mechanics and black holes and going to more simple terms of entropy (because simple to more complex is a boring way to teach). Imagine a drop of ink and a bowl of water, if we were to drop the ink into the water then it would spread until it was equally distributed throughout the water (maximum entropy). We would still have the same amount of ink throughout the entire bowl, but it would have mixed with the water. Entropy increases in the way that ink will only spread through the water, but will not slowly un-mix back into the drop that first entered the water.