What is a jet engine's fuel flow at constant IAS?
The most obvious difference is due to the temperature of the air.

Both turbine and piston engines are heat engines. They work by converting thermal power into mechanical power. The theoretical absolute maximum efficiency you can achieve is called the Carnot efficiency, $$\eta = 1-\dfrac{T_C}{T_H}$$

This is the efficiency of an ideal engine using the Carnot cycle, that works by transferring heat from a hot reservoir with temperature $T_H$ to a cold reservoir with temperature $T_C$. A typical jet engine is approximated by the Brayton cycle, and a piston engine the Otto or Diesel cycle, but neither can ever by more efficient than the efficiency noted above. The efficiency of a Brayton cycle is $$\eta = 1-\dfrac{T_C}{T_E}$$ with $T_E$ the EGT.

When flying higher, the temperature of the cold reservoir (the atmosphere) drops lower. You can see that the maximum efficiency of the engine will also increase (even if $T_H$ or $T_E$ respectively drop simultaneously with $T_C$). This means that, even if the required power would stay constant for constant thrust, the fuel flow changes, because a single unit of energy from a drop of fuel can be converted into more mechanical power.