2D halide perovskites are among intensely studied materials platforms profiting from solution-based growth and chemical flexibility. They feature exceptionally strong interactions among electronic, optical, as well as vibrational excitations and hold a great potential for future optoelectronic applications. A key feature for these materials is the occurrence of structural phase transitions that can impact their functional properties, including the electronic band gap and optical response dominated by excitons. However, to what extent the phase transitions in 2D perovskites alter the fundamental exciton properties remains barely explored so far. Here, we study the influence of the phase transition on both exciton binding energy and exciton diffusion, demonstrating their robust nature across the phase transition. These findings are unexpected in view of the associated substantial changes of the free carrier masses, strongly contrast broadly considered effective mass and drift-diffusion transport mechanisms, highlighting the unusual nature of excitons in 2D perovskites.