This question is somewhat nonsensical because it is current, not power, that maintains the magnetic field in a solenoid. Also, you haven't defined P, so we obviously can't say if it changes. What happens in the circuit also depends on how the solenoid is driven, like with constant voltage, constant current, or something else.
When you are driving a solenoid in steady state (DC), the amount of energy stored in its field is irrelevant to the circuit. All the driving circuit sees is the resistance of the coil. Hence, the steady state voltage and current will be the same before and after the core is changed.
Adding a high permiability material into the core increases the total energy stored there for the same ampere-turns. This means a temporary blip of energy must be added, which will show up, for example, as increased voltage if the current is held constant.