We study the cosmological properties of a codimension two brane world that sits at the intersection between two four-branes, in the framework of six-dimensional Einstein–Gauss–Bonnet gravity. Due to contributions of the Gauss–Bonnet terms, the junction conditions require the presence of localized energy density on the codimension two defect. The induced metric on this surface assumes a FRW form, with a scale factor associated with the position of the brane in the background; we can embed in the codimension two defect the preferred form of energy density. We present the cosmological evolution equations for the three-brane, showing that, for the case of pure AdS₆ backgrounds, they acquire the same form as the ones for the Randall–Sundrum II model. When the background is different from pure AdS₆, the cosmological behaviour is potentially modified in respect to the typical one of codimension one brane worlds. We discuss, in a particular model embedded in an AdS₆ black hole, the conditions one should satisfy in order to obtain standard cosmology at late epochs.