Energy-dependent patterns in the arrival directions of cosmic rays are searched for using data of the Pierre Auger Observatory. We investigate local regions around the highest-energy cosmic rays with E > = 6×1019 eV by analyzing cosmic rays with energies above E > = 5×1018 eV arriving within an angular separation of approximately 15°. We characterize the energy distributions inside these regions by two independent methods, one searching for angular dependence of energy-energy correlations and one searching for collimation of energy along the local system of principal axes of the energy distribution. No significant patterns are found with this analysis. The comparison of these measurements with astrophysical scenarios can therefore be used to obtain constraints on related model parameters such as strength of cosmic-ray deflection and density of point sources.