High-temperature latent heat thermal energy storage with metallic alloy phase change materials (PCMs) utilize the high latent heat and high thermal conductivity to gain a competitive edge over existing sensible and latent storage technologies. Novel macroporous latent heat storage units can be used to enhance the limiting convective heat transfer between the heat transfer fluid and the PCM to attain higher power density while maintaining high energy density. 3D monolithic percolating macroporous latent heat storage unit cells with random and ordered substructure topology were created using synthetic tomography data. Full 3D thermal computational fluid dynamics (CFD) simulations with phase change modeling was performed on 1000+ such structures using effective heat capacity method and temperature- and phase-dependent thermophysical properties. Design parameters, including transient thermal and flow characteristics, phase change time and pressure drop, were extracted as output scalars from the simulated charging process. As such structures cannot be parametrized meaningfully, a mesh-based Geodesic Convolutional Neural Network (GCNN) designed to perform direct convolutions on the surface and volume meshes of the macroporous structures was trained to predict the output scalars along with pressure, temperature, velocity distributions in the volume, and surface distributions of heat flux and shear stress. An Artificial Neural Network (ANN) using macroscopic properties—porosity, surface area, and two-point surface-void correlation functions—of the structures as inputs was used as a standard regressor for comparison. The GCNN exhibited high prediction accuracy of the scalars, outperforming the ANN and linear/exponential fits, owing to the disentangling property of GCNNs where predictions were improved by the introduction of correlated surface and volume fields. The trained GCNN behaves as a coupled CFD-heat transfer emulator predicting the volumetric distribution of temperature, pressure, velocity fields, and heat flux and shear stress distributions at the PCM–HTF interface. This deep learning based methodology offers a unique, generalized, and computationally competitive way to quickly predict phase change behavior of high power density macroporous structures in a few seconds and has the potential to optimize the percolating macroporous unit cells to application specific requirements.