Increasingly commonplace in cities, extreme heat events introduce multi-stress vulnerability, affecting people's health and well-being, financial situation, mobility, social relations, and access to basic services. Planning to reduce heat vulnerability has become part of government business and to some extent community-level responses, cutting across a number of sectors including public health, emergency management, social services, critical infrastructure, and housing. This planning is often framed around heat as an emergency, focusing on preventing loss of life and severe health impacts, yet a vulnerability perspective also draws attention to the chronic and persistent impacts of heat. Our research, based on interviews and desktop research in Melbourne, Australia, found tensions between addressing heat as an emergency and heat as a source of chronic stress, with emergency responses taking precedence over responses addressing the chronic dimensions of heat. Each approach results in different but nonetheless related programmatic priorities for reducing vulnerability. In complex institutional settings, improving relations between policy and programme managers, non-government organisations, and vulnerable people themselves would enable the multiple stresses associated with extreme heat to be more effectively addressed. Policy and institutional responses that better appreciate the interconnections between the emergency and chronic aspects of heat would likely reduce vulnerability and contribute to more just approaches to urban sustainability.