Sure, it varies. For example, molar heat capacity at constant pressure of iron changes from 21,59 J/(molK) at 200 K to 2,15 J/(molK) at 300 K, and to 32,05 J/(molK) at 600 K, according to CRC Handbook of Chemistry and Physics.
The variation of specific heat with temperature for different steels can also be found in the book entitled "Heat Treater's Guide" published by ASM International.
Yes, the heat specific to steel depends on temperature, the matter is compiled by phase transitions taking place in steel, which introduce disturbances related to the heat of phase transitions. Dhanaji Mahadeo Kale Please take a look at this article: