The presence of repeatable behaviours that vary between individuals (animal personality) and the mechanisms that underpin these behaviours are of significant ecological and evolutionary importance. With current climatic trends predicted to facilitate biological invasions, we sought to examine how personality and behavioural syndromes may influence invasion potential in the face of a changing climate. We assessed a population of an invasive crayfish, Cherax destructor, in a laboratory setting for (1) the presence of temporally and contextually repeatable behavioural differences between individuals along three axes: (i) activity, (ii) boldness/shyness and (iii) exploration/avoidance, (2) correlations between body size, sex and behavioural traits at the between-individual level, (3) the influence of predicted near-future increases in temperature on personality and (4) the presence of behavioural syndromes (between-individual correlations among behaviours). Individuals exhibited repeatable variation along the three behavioural axes, providing support for the presence of animal personality. Body size was negatively correlated with exploration and activity, but sex was unrelated to personality across the three axes. Exploration increased with temperature, whereas the effect of temperature on activity was dependent on the treatment order. Finally, activity and exploration were positively correlated, indicative of a behavioural syndrome. This study provides evidence of state-dependent personality and support for the existence of behavioural syndromes which may facilitate biological invasions. Our work also provides evidence to suggest that predicted future increases in temperature will promote exploratory behaviour in invasive populations of C. destructor.