The Firefighter Problem is a single-player game in which a fire (modelling a rumour, disease, or other contagion) spreads through a graph and we are tasked with defending nodes to save some target number of them. This problem is formally computationally hard in general, but tractable in some restricted cases. We introduce a Firefighter variant called The Cost Function Firefighter Problem , in which each node has a cost to defend that can vary depending on time and the burning state of the graph - for example, as a function of proximity to fire (which we call threat). We show that The Cost Function Firefighter Problem is formally computationally hard, even on a class of trees where classic firefighter is tractable. We complement our theoretical results with an empirical investigation of the performance of threat and degree-based heuristics when our cost functions are threat-based or uniformly random, finding that the relative performance of heuristics is dependent on the class of random graph the process is run on.