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High-temperature heat pumps (HTHPs) are becoming increasingly relevant in the industry as they represent a promising solution for decarbonising industrial heat. These technologies can enable the electrification of industrial processes by exploiting electricity from renewables to provide process heat at temperatures above 250 °C, as in the case of emerging Brayton-based HTHPs. To succeed in this purpose, HTHPs must also ensure operational flexibility, which entails the ability to operate safely under varying loads and promptly respond to fluctuations in demand, while maintaining high efficiencies. Moreover, the ability to provide large flexible electric loads to transmission system operators, has the potential to unlock innovative business cases and further promote the use of these systems. Common control strategies for achieving this, include employing bypass mechanisms, fluid inventory control, and adjusting turbomachinery rotational speeds. Despite their variety, the simultaneous use of such control strategies is often limited as they may lead to significantly different system behaviours, both in terms of transient and steady performance. In this paper, rotational speed and fluid inventory control are examined from a transient perspective to maintain the desired sink temperature while regulating the thermal load of the system. Results indicate that rotational speed control leads to negligible sink temperature residuals, while fluid inventory control better preserves the HTHP performances for varying temperature glides.