Consider a rotating object, If its moment of inertia decreases its angular velocity increases, in order to conserve angular momentum. But what happens to rotational kinetic energy? Why it changes, we are not applying any force/ torque. Neglect all types of frictions, and take an ideal case.
Answer
As Bill N indicated, there does have to be a force in order to rearrange the particles in the system so that the moment of inertia changes. This means that work is being done on the system (or extracted from the system), so that the total energy can change.
You also suggested that a rotating body might change size due to temperature changes, but this answers your own question: changing the temperature would require a change in the energy of the system.
Finally, you asked how a force could change the moment of inertia without affecting the angular momentum. If the force is directed radially, it does not create a torque on the system so there would be no change in angular momentum.
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