Mixed-criticality (MC) systems have recently been devised to address the requirements of real-time systems in industrial applications, where the system runs tasks with different criticality levels on a single platform. In some workloads, a high-critically task might overrun and overload the system, or a fault can occur during the execution. However, these systems must be fault tolerant and guarantee the correct execution of all high-criticality (HC) tasks by their deadlines to avoid catastrophic consequences, in any situation. Furthermore, in these MC systems, the peak-power consumption of the system may increase, especially in an overload situation and exceed the processor thermal design power (TDP) constraint. This may cause generating heat beyond the cooling capacity, resulting the system stop to avoid excessive heat and halting the processor. In this article, we propose a technique for dependent dual-criticality tasks in fault-tolerant multicore MC systems to manage peak-power consumption and temperature. The technique develops a tree of possible task mapping and scheduling at design-time to cover all possible scenarios and reduce the low-criticality task drop rate in the HC mode. At the runtime, the system exploits the tree to select a proper schedule according to fault occurrences and criticality mode changes. Experimental results show that the average task schedulability is 74.14% on average for the proposed method, while the peak-power consumption and maximum temperature are improved by 16.65% and 14.9 °C on average, respectively, compared to a recent work. In addition, for a real-life application, our method reduces the peak power and maximum temperature by up to 20.06% and 5 °C, respectively, compared to a state-of-the-art approach.