Tackling transient conjugate heat transfer with high-fidelity methods such as large-eddy simulation (LES) requires to couple the LES solver with a heat transfer solver in the solid parts of the computational domain. Challenges include performance scalability, numerical stability and accuracy. In such unsteady simulations, both solvers integrate their respective set of equations in time independently for the sake of computational efficiency, and during a physical time corresponding to a coupling period to be specified. During the separate temporal integrations, the thermal state at the wall interface is typically set as a Dirichlet condition in the flow solver while a Neumann condition is imposed in the heat transfer solver to enhance numerical stability. When carefully validated, the chosen value of the coupling period which optimal value is initially unknown can be compared a posteriori with a refined solution. However, this optimal value of the coupling period (neither too large to remain accurate nor too short not to penalize the computational cost) is case-dependent. In this study, an approach to automatically adapt the coupling period is presented. It relies on a describing the temporal evolution of the boundary temperature in hybrid cells composed of the neighboring fluid and solid mesh cells. Then, between coupling iterations, each solver advances separately with the same Dirichlet boundary condition on the computed interface temperature. Yielding a first order Ordinary Differential Equation (ODE) for the boundary temperature, the method allows using automatic adaptation of the step size to control the numerical integration error based on a prescribed tolerance by using controllers. The coupling method is studied on 1D unsteady configurations where the results demonstrate that this energy conserving method is able to determine the coupling period automatically and efficiently for different configurations. The impact of excitation frequency and prescribed tolerance enables to select a specific PID controller which remains robust in spite of not carrying out step rejections for the sake of computational performance in the context of an LES application.