The energy transition leads to a substantial increase in the amount of power that must be transported through medium-voltage electricity grids. Cable capacity is constrained by the maximum permissible cable temperature. Therefore, Dutch distribution system operators require real-time insight into cable temperatures to enable optimal utilization of grid capacity. We compute cable temperatures by solving the heat equation subject to realistic time dependent forcing, using the finite element method with implicit Euler time stepping. A major challenge in estimating cable temperatures is dealing with uncertainties in soil properties and power forecasts. Monte Carlo sampling is applied to to quantify the reliability of model predictions under this inherent parameter uncertainty. This is expensive in terms of computing time. To reduce computational cost while maintaining accuracy, we employ both Multilevel Monte Carlo (MLMC) and Multi-Index Monte Carlo (MIMC) techniques separately, combined with adaptive mesh refinement in the spatial domain in which the a posteriori error is controlled. Using these techniques we see a significant reduction in computational time. To our knowledge, the application of these methods to cable-temperature modeling is novel, and the results demonstrate the potential of MIMC for parabolic partial differential equations.