Although atomic clocks have very low levels of frequency instability, they are nonetheless sensitive (albeit slightly) to various environmental parameters, including temperature, power supply voltage, and dc magnetic fields. In the terrestrial environment, however, atmospheric pressure (i.e., the air's molecular density) is not generally included in this list, because the air's density variations near the surface of the earth will typically have a negligible effect on the clock's performance. The situation is different, however, for clocks onboard satellites like Galileo, where manufacturing and testing are done at atmospheric pressure, while operation is in vacuum. The pressure sensitivity of atomic clocks, in particular vapor-cell atomic clocks, can therefore be of significance. Here, we discuss some of the ways in which changes in atmospheric pressure affect vapor-cell atomic clocks, and we demonstrate that, for one device, the pressure-sensitivity traces back to a pressure-induced change in the temperature of the clock's filter and resonance cells.