Topology optimised designs for passive cooling of light-emitting diode (LED) lamps are investigated through extensive numerical parameter studies. The designs are optimised for either horizontal or vertical orientations and are compared to a lattice-fin design as well as a simple parameter optimised commercial pin fin design. The different orientations result in significant differences in topologies. The optimisation favours placing material at outer boundaries of the design domain, leaving a hollow core that allows the buoyancy forces to accelerate the air to higher speeds. Investigations show that increasing design symmetry yields performance with less sensitivity to orientation with a minor loss in mean performance. The topology-optimised designs of heat sinks for natural convection yield a 26% lower package temperature using around 12% less material compared to the lattice-fin design, while maintaining low sensitivity to orientation. Furthermore, they exhibit several defining features and provide insight and general guidelines for the design of passive coolers for LED lamps. (C) 2018 Elsevier Ltd. All rights reserved.