ABSTRACT The nature of the shallow decay phase in the X-ray afterglow of the gamma-ray burst (GRB) is not yet clarified. We analyse the data of early X-ray afterglows of 26 GRBs triggered by Burst Alert Telescope onboard Neil Gehrels Swift Observatory and subsequently detected by Fermi Large Area Telescope (LAT) and/or Imaging Atmospheric Cherenkov Telescopes. It is found that nine events (including two out of three very-high-energy gamma-ray events) have no shallow decay phase and that their X-ray afterglow light curves are well described by single power-law model except for the jet break at later epoch. The rest are fitted by double power-law model and have a break in the early epoch (around ks), however, eight events (including a very-high-energy gamma-ray event) have the pre-break decay index larger than 0.7. We also analyse the data of well-sampled X-ray afterglows of GRBs without LAT detection and compare their decay properties with those of high-energy and very-high-energy gamma-ray events. It is found that for the GeV/TeV bursts, the fraction of events whose X-ray afterglows are described by single power law is significantly larger than those for non-GeV/TeV GRBs. Even if the GeV/TeV GRBs have shallow decay phase, their decay slope tends to be steeper than non-GeV/TeV bursts, that is, they have less noticeable shallow decay phase in the early X-ray afterglow. A possible interpretation along with the energy injection model is briefly discussed.