High-temperature heat pumps represent a valuable technology to decrease fossil fuel consumption in industry, as they can use renewable electricity to cover a given heat demand. Conventional heat pumps provide heat at temperatures around a maximum of 80 °C. Nowadays, heat production up to 150 °C can be achieved with high-temperature heat pumps. For higher temperatures between 150 °C and 250 °C, specialized designs, such as Brayton heat pumps, are required. This paper aims to investigate the transient response of the DLR's CoBra prototype, an innovative Brayton-cycle heat pump intended to provide heat above 250 °C and currently under commissioning at the DLR facility in Cottbus, Germany. First, a comprehensive transient thermodynamic model of the system is developed, accounting for heat exchangers and piping thermal inertia. Furthermore, a control logic is presented that ensures safe operation throughout off-design conditions and start-up manoeuvres. In particular, several control parameters are considered to avoid potential operational issues, such as critical temperature gradients, compressor surge, and critical mechanical vibration phenomena due to resonance. The performed simulations aim to reduce start-up time and energy consumed during start-up. Results show that with the help of the described controller, the system can reach design operation via a transient trajectory safely and quickly. Therefore, the capability of the CoBra prototype to flexibly supply high-temperature heat is demonstrated.