This paper presents for the first time an approach to minimize direct operational costs (DOC) for all-electric aircraft during the climb phase, introducing a time-varying cost index (CI). The CI is modeled as a dynamic parameter commanded by Air Traffic Control (ATC), allowing the aircraft to maintain a constant airspeed throughout the climb, while respecting the air traffic regulations. This paper also explores the implications of a time-varying CI on the determination of optimal airspeed and climbing time for all-electric aircraft. Additionally, it provides the necessary equations to calculate both the optimal climb airspeed and climb duration. The proposed methodology has been validated through a simulated scenario that reflects actual operational procedures. As a result, optimal values for climb airspeed, climbing time, and energy consumption have been established, paving the way for future applications of this methodology to advanced air mobility all-electric vehicles.