Embedded systems are vulnerable to significant security threats from Fault Injection Attacks (FIAs), which allow attackers to gain access to confidential information. While various attack detectors have been proposed in the literature to detect different types of FIAs, these detectors themselves are susceptible to such attacks and can be compromised. Hence, the robustness of these detectors is critical in maintaining the security of embedded systems. The focus of this study is to evaluate the robustness of digital circuits and delay-based digital detectors against a new type of FIA called Power-Off Attack (POA). POA occurs when the power to the chip is turned off, and the detectors are not active. Following a POA attack, the circuit or its detectors may not function properly when the power is turned back on, which can allow other attacks to be applied without being detected if the detectors are less sensitive. This study implements two detectors on Xilinx Artix-7 FPGAs and examines the impact of heating cycles on detector characteristics when the FPGA is in various states, including power-off, power-on, and inactive states (such as clock-freezing mode). Our experiments reveal that heating cycles in power-off mode can alter the FPGA component delays and the accuracy of its detectors, which highlights the vulnerability of these systems to POA and potential issues for embedded system security.