It is known that the critical properties of the adsorption transition for the self-avoiding walk model of lattice polymers are not superuniversal. In particular this is because the value of the crossover exponent \(\phi\) in the three-dimensional case deviates from the mean-field prediction \(\phi\)  = 1/2. The mean-field value is exactly true for high-dimensional lattices with dimension d > 4, where d = 4 is the upper critical dimension. These high-dimensional cases have not been studied extensively and here we employ Monte Carlo simulations to investigate the scaling of the adsorbed fraction, finding that the expected mean-field scaling holds for d = 5, 6. However, we do not find additional logarithmic scaling for d = 4, contrary to what is typically expected for lattice models at the upper critical dimension.