Refractory metal high-entropy superalloys (RSA), which possess a nanoscale microstructure of B2 and bcc phases, have been developed to offer high temperature capabilities beyond conventional Ni-based alloys. Despite showing a number of excellent attributes, to date there has been little consideration of their microstructural stability, which is an essential feature of any material employed in high temperature service. Here, the stability of the exemplar RSA AlMo0.5NbTa0.5TiZr is studied following 1000 h exposures at 1200, 1000 and 800 °C. Crucially, the initial nanoscale cuboidal B2 + bcc microstructure was found to be unstable following the thermal exposures. Extensive intragranular precipitation of a hexagonal Al-Zr-rich intermetallic occurred at all temperatures and, where present, the bcc and B2 phases had coarsened and changed morphology. This microstructural evolution will concomitantly change both the mechanical and environmental properties and is likely to be detrimental to the in-service performance of the alloy.