The discovery of planets orbiting double stars at close distances has sparked increasing scientific interest in determining whether Earth-analogues can remain habitable in such environments and how their atmospheric dynamics is influenced by the rapidly changing insolation. In this work we present results of the first three-dimensional numerical experiments of a water-rich planet orbiting a double star. We find that the periodic forcing of the atmosphere has a noticeable impact on the planet's climate. Signatures of the forcing frequencies related to the planet's as well as to the binary's orbital periods are present in a variety of climate indicators such as temperature and precipitation, making the interpretation of potential observables challenging. However, for Earth-like greenhouse gas concentrations, the variable forcing does not change the range of insolation values allowing for habitable climates substantially.