We propose the extended minimal state cell (EMSC), a recurrent neural network cell tailored for temperature- and rate-dependent constitutive modeling. In contrast to standard RNNs, the EMSC is specifically designed to ensure stationarity and self-consistency. This makes it particularly well-suited for numerical simulations as it provides stable predictions and convergence with refined input discretization. Based on these conditions, we derive a general framework for the EMSC transition functions and a custom loss definition. Similar to the rate-independent MSC model, the EMSC decouples the number of state variables from the number of trainable parameters. This allows us to create neural network models with a minimal number of state variables for high physical interpretability. The EMSC is trained and validated based on random walk datasets of eight distinct materials. They comprise one-dimensional rheological models, advanced three-dimensional thermo-visco-plasticity theories as well as heterogeneous materials, i.e. a four-ply laminate and a rigid particle reinforced rubber. We demonstrate that EMSC models provide accurate predictions of the large deformation responses of all materials probed. The trained models require less than 25000 parameters, while maintaining the same number of state variables as their physics-based counterparts. Additionally, we show that the EMSC model trained on random walks generalizes to different input distributions, like spline-based trajectories or inputs with constant rates. In addition to prediction accuracy, we evaluate the self-consistency of the trained models and compare it with the results of a rate-dependent LSTM model. While the EMSC yields self-consistent predictions for all materials, the performance of the LSTM model rapidly declines as the input trajectories are refined. Due to its minimal state space, compact parameter space, self-consistency, and high expressivity, the EMSC is a promising candidate for surrogate modeling, particularly for materials for which reliable micromechanical models are available to generate rich training data.