When the tea in an open teacup reaches an equilibrium temperature with its surroundings, it stops cooling.

Heat flows from a body at a higher temperature to another body at a lower temperature. The net exchange of heat is zero when two bodies are at the same temperature.

In our case, when tea in an open teacup is hot (higher temperature), heat flows from it to its surroundings (lower temperature). This process continues until the temperature of the tea in the teacup matches the temperature of the surroundings. Thereafter, it does not lose its energy. Therefore, its temperature does not change. And, we observe that it has stopped cooling.

Hence, tea in an open teacup stops cooling after some time.