A question on superconductivity

Let us suppose that there is a superconducting toroid. Let us also suppose that there is a finite electric current flowing in it. We imagine the temperature of the toroid to be below the critical temperature. Now if we try to raise the temperature of the toroid then due to the law of conservation of energy (or lenz’s law) the magnetic field which is due to the circulating current would resist to being ‘changed’ easily. This would lead to the circulating current trying to remain ‘unchanged’ which would only be possible if the temperature of the superconductor is prevented from going above the critical temperature. So there might arise two possibilities (according to my understanding)
1) There would exist some kind of a thermal inertia in this case which would result in the specific heat capacity of the superconductor being bumped up at the interval T(0-)->T(0+) [T being the critical temperature of the super conductor]
2) The critical temperature of a current carrying superconductor is ‘raised’ while a transition from ‘below critical’ temperature to ‘above critical’ temperature is tried.

Does any of these things actually happen? I am curious.

DrDu
I don't see the problem. If the temperature is rised, the material becomes a normal conductor and the current in the first moment will be the same as in the superconducting state. However, Ohmian resistance will soon reduce the current. The amount of heat generated will equal the energy stored initially in the magnetic field.

when the temperature rises from below the critical temperature point- nothing happens to the current's magnitude until the temperature reaches the critical point. At T->T(0+) the resistance of the super conductor suddenly becomes a finite positive quantity from zero. that's like a step impulse. The current too therefore should be prone to changing all of a sudden in magnitude. This sudden-ness is what IMHO lenz's law wouldn't let happen.

DrDu