As IoT in a multi-server environment increases resources’ utilization, more and more problems of IoT authentication and key agreement are being revealed. The Authentication and Key Agreement (AKA) protocol plays an important role in solving these problems. Many AKA protocols have been proposed, and some of them support their own verifications. However, a unifying verification framework for multi-server IoT is lacking. In this article, we propose a formal verification framework of AKA protocols for multi-server IoT (FVF-AKA). It supports the construction of CSP models for the AKA protocol, the implementation of the CSP models in PAT with C#, and the verification of formal models. With the help of C#, many complex functions in the AKA protocol can be implemented. We also design an algorithm to support automatic conversion from the CSP model to the PAT model. FVF-AKA can verify four fundamental properties (deadlock freedom, entity legitimacy, timeout delay, and session key consistency). It also supports the verification of security properties for the AKA protocol suffering from four different attacks (relay attacks, denial of service attacks, server spoofing attacks, and session key attacks). Our approach can be applied to most AKA protocols for multi-server IoT generally. By applying FVF-AKA to two AKA protocols, we can verify whether they satisfy the fundamental properties and analyze their security properties in vulnerable environments. Our work would help to analyze the AKA protocol for multi-server IoT and provide the foundation for the analysis of enhancing its security and robustness.