Rotational energy of stellar remnants The theoretical maximum rotational energy a black-hole can have is 29% of its rest mass. We've often observed remnant black-holes spinning at nearly the speed of light. From this we conclude that remnant black-holes weighing in at several solar masses often have more than a solar-mass of their mass-energy stored in their spin. However, this is an insane amount of energy, and it doesn't seem reasonable that original stars could have so much spin-energy.
Keeping conservation of energy in mind, did the original stars have that much rotational energy? If not, where is the energy coming from? Is there some auxiliary process that generates the incredible amount of spin-energy found in stellar remnants?
 A: The original energy isn't necessarily in the form of rotational kinetic energy.
To address this let me first in general go over the process of increase of angular velocity upon contraction of a rotating system.
The simplest case is two objects, connected with an infinitely strong cable, orbiting their common center of mass, and machinery is reeling in that cable. We assume that this contracting machinery has an astronomical capability to increase exerted force to provide whatever force is required to maintain contraction.
As cable is reeled in the motion of the objects isn't circular motion; the objects move along an inward spiral. All during this process the centripetal force exerted by the contracting machinery can be decomposed in the following two components:
-perpendicular to the instantaneous velocity
-parallel to the instantaneous velocity
The component parallel to the instantaneous velocity is causing angular acceleration. So that centripetal force is doing work.
The faster the objects are orbiting around their common center of mass the bigger the required centripetal force.
As long as the machinery can keep ramping up the exerted force the rotating system can keep contracting.

The above shows the energy source that powers the increase of rotational kinetic energy of a contracting star. Gravitational potential energy is converted to kinetic energy. As we know: the more a star contracts the greather the density, the stronger the contracting gravitational force.

Stars below a particular mass limit end up as a white dwarf. All of the energy radiated during its life came from gravitational potential energy being released as the star was gravitationally contracting to the final white dwarf diameter. The temperature required for nuclear fusion to happen was reached by release of gravitational potential energy upon contraction. The final white dwarf state leaves an astronomical amount of gravitational potential energy unused. As we know, degeneracy pressure keeps the star from contracting further, preventing further release of gravitational potential energy.
Above a particular mass limit the gravitational contraction overpowers the degeneracy pressure. When that happens the amount of gravitational potential energy that is subsequently released is larger than all that was released during the entire luminous lifetime of the star.



I'm guessing that you were under the impression that the rotational kinetic energy of any rotating system is conserved, just as the angular momentum is conserved.
The following cannot be emphasized enough: the process of angular acceleration due to contraction of a rotating system is a process of energy conversion. The centripetal force is doing work; potential energy is converted to kinetic energy.


In response to a comment:
Linear momentum and angular momentum have in common that there are cases where momentum considerations allow a faster calculation.
Linear momentum:
A cannon fires a projectile. The explosion of the artillery propellent causes two things: recoil of the cannon, and the projectile shoots out of the barrel. Given are the mass ratio of cannon and projectile, and the velocity of the recoil. Then the kinetic energy of the projectile can be calculated as follows: the velocity of the recoil is known; from conservation of momentum infer the velocity of the projectile; from that infer the kinetic energy of the projectile. You then find (perhaps against expectation), that the kinetic energy of the projectile is larger than that of the recoil.
The cause of the kinetic energy of the projectile is the explosion of the artillery propellent. It would be wrong to suggest: the projectile shoots out of the barrel because the cannon recoils.
Causality is associated with things that happen sequentially in time; potential energy transformed to kinetic energy. Conservation of momentum is a spatial principle.
Angular momentum
It should be possible to calculate the increase of rotational kinetic energy by integrating the work done. But it's tricky; if the calculation is in terms of doing work against a centrifugal potential: the work that the centripetal force is doing is increasing that very centrifugal potential.
As we know: there is an impressively faster way: use conservation of angular momentum to infer how much the angular velocity increases upon contraction, and infer the increase of rotational kinetic energy from that.
However, the fact that angular momentum offers the smoother calculation does not mean that the angular momentum conservation is the cause of the angular acceleration. Conservation of angular momentum is a spatial principle
A: Let us assume the remnant, of mass $M$, was the core of a more massive star. Let the initial rotation rate of that core (assume a uniform solid body for simplicity) be $\omega_i$ and the initial moment of inertia be $2MR^2/5$, where $R$ would be the core radius.
Now allow it to collapse to a much smaller radius $r$ whilst conserving the core mass and angular momentum and examine the ratio of rotational kinetic energy to rest mass energy, which I will label as $\alpha$.
Before collapse:
$$\alpha_i =\frac{R^2\omega_i^2}{5c^2}$$
After collapse:
Assuming a homologous collapse and conservation of angular momentum, the new (final) angular velocity will be
$$\omega_f = \omega_i \frac{R^2}{r^2}\, ,$$
with a final moment of intertia of $2Mr^2/5$. The ratio of rotational to rest mass energy is
$$ \alpha_f = \frac{r^2\omega_f^2}{5c^2} = \frac{r^2 \omega_i^2}{5c^2} \left(\frac{R^4}{r^4}\right) = \alpha_i\left(\frac{R^2}{r^2}\right)\ .$$
Thus the ratio of rotational to rest mass energy can become arbitrarily large as $r \rightarrow 0$.
Note that "conservation of rotational energy" is not a thing. The law is: conservation of total energy. That energy can be shared out between different forms in different ways. In this case, energy can be conserved by switching some of the gravitational potential energy released in the collapse to rotational kinetic energy.
The trick is actually to work out how to shed enough angular momentum to allow the black hole to form in the first place.
The rotational energy grows as $r^{-2}$ whilst gravitational potential energy goes as $r^{-1}$. There will be a minimum radius where the collapsing object becomes rotationally supported. It turns out that is much larger than the Schwarzschild radius, so angular momentum must be lost to allow the formation of a black hole.
