Is the rate constant over time or does it vary over time?

Theoretically, yes it is constant, but sometimes it can vary from one isotope to another.
No, it is not constant at all

It depends on the specific reaction and conditions. In general, the rate constant can be considered constant if the reaction conditions remain unchanged. However, certain factors such as temperature, concentration, presence of catalysts, and reaction mechanisms can cause the rate constant to vary over time. Therefore, in practice, the rate constant can be both constant and variable depending on the circumstances.