Thanks to their capability to provide very large mode area together with effective suppression of high-order modes, while allowing strong pump absorption and efficient conversion, Yb-doped double-cladding photonic crystal fibers are one of the key enabling factors for the development of high power fiber lasers. Thermal effects are currently appointed as the main bottleneck for future power scaling since, beyond a certain average power, they allow guidance of high order modes and energy transfer to them, causing a sudden degradation of the beam quality. In this paper the effects of heat load on the modes of double cladding fibers are thoroughly analyzed with a full-vector modal solver based on the finite-element method with integrated steady-state heat equation solver. Fibers with different inner cladding designs are compared to provide a deeper understanding of the mechanisms beyond the mode reconfinement and coupling. The influence of the fiber design on the robustness of the single-mode regime with respect to fiber heating has been demonstrated, providing a clear picture of the complex interaction between modes. On the basis of simulation results it has been possible to group fiber modes into three families characterized by peculiar reaction to heating.