Context. One science goal of the upcoming Planck mission is to perform a full-sky cluster survey based on the Sunyaev-Zel'dovich (SZ) effect, which leads to the question of how such a survey would be affected by cosmological models with a different history of structure formation than Lambda CDM. One class of these models are early dark energy (EDE) cosmologies, where the dark energy contribution does not vanish at early times. Aims. Since structures grow slower in the presence of EDE, one expects an increase in the number of galaxy clusters compared to Lambda CDM at intermediate and high redshifts, which could explain the reported excess of the angular CMB power spectrum on cluster scales via an enhanced SZ contribution. We study the impact of EDE on Planck's expected cluster sample. Methods. To obtain realistic simulations, we constructed full-sky SZ maps for EDE and Lambda CDM cosmologies, taking angular cluster correlation into account. Using these maps, we simulated Planck observations with and without Galactic foregrounds and fed the results into our filter pipeline based on the spherical multi-frequency matched filters. Results. For the case of EDE cosmologies, we clearly find an increase in the detected number of clusters compared to the fiducial Lambda CDM case. This shows that the spherical multi-frequency matched filter is sensitive enough to find deviations from the Lambda CDM sample, being caused by EDE. In addition we find an interesting effect of EDE on the completeness of the cluster sample, such that EDE helps to obtain cleaner samples.