Cosmology presents students with ideas that stimulate their curiosity and brings together various concepts from STEM that call on a variety of reasoning types across multiple representational modes, involving subtleties of spacetime relations, a variety of models and evidence requiring multiple lines of high precision observations. This study investigated high school students' levels and types of reasoning that frame their conceptions in different cosmology topics. An open-ended knowledge survey, the Cosmology Knowledge Survey (CosmoKS), was developed and implemented online to 286 high school students (aged 16-18 years) from Australia and Sweden. A modified version of the Structure of the Observed Learning Outcome (SOLO) taxonomy with four levels (pre-structural, uni-structural, multi-structural and relational) was used as a guide to analyse students' open-ended and structured responses. This provided insights into the level and complexity of reasoning underpinning a variety of conceptions across the four dimensions of cosmology education: size and scale, spacetime location, composition of the universe and evolution of the universe. The study identified underlying patterns in student reasoning and conceptions in cosmology, summarised as (i) navigating spatial and temporal relations, (ii) counterintuitive concepts and (iii) language and everyday experience, especially intuition. The analysis led to the characterisation of a hierarchy of reasoning that helps identify sources of alternative conceptions and provided the basis for the development of a concept inventory and progression with broad implications.