When taking into account the generalized uncertainty principle in statistical physics, the density of states must make a correction, which causes all the results of traditional statistical physics to have different degrees of correction. In high-energy or high-temperature conditions, this amendment can subvert the traditional concept and there are also some certain amendments at low temperatures. In this paper we study the thermodynamic properties of the ideal and weakly interacting Fermi gas in low temperature conditions when the generalized uncertainty principle is taken into account. Firstly, analytical expressions of chemical potential, internal energy and heat capacity at constant volume of ideal or weakly interacting Fermi gas are given. Then the properties of copper electron gas are computed as an example, showing that when the generalized uncertainty principle is taken into account the chemical potential, Fermi energy and the ground state energy increase with the increase of temperature, while the heat capacity decreases. When the temperature is lower than 0.3 times TF0, the internal energy increases with the increase of temperature, but becomes decreased when temperature is high than 0.3 times TF0. These amendments are mostly dependent on particle density, which becomes bigger and bigger with particle density increasing.