Ni-Mn-Ga ferromagnetic shape memory alloys (FSMAs) have attracted great attention for more than two decades, due to their large magnetic shape memory effect that originates from the rearrangement of martensitic variants under an external magnetic field. Over the past decade, accumulated knowledge on the properties of Ni-Mn-Ga Heusler alloys has allowed people to foresee the possibility of employing these alloys in device applications. However, the low operating temperatures and high brittleness remain the major drawbacks for the industrial application. Consequently, there has been growing interest in the modification of Ni-Mn-Ga alloys by adding a fourth element to increase transformation temperatures and to improve ductility. A recent study shows that the ductility has been effectively improved in Cu-doped Ni-Mn-Ga alloy under the situation of single phase via strengthening grain boundaries. In addition, the crystal structure, martensitic transformation, magnetic properties, high temperature magnetoplasticity and magnetocaloric effect have been reported in Ni-Mn-Ga-Cu alloys. Experimental results have shown that the martensitic transformation temperature (T_m) is drastically increased and the Curie temperature (T_C) slightly decreased with Cu addition. As already known, the alloying elements affect both the crystal and electronic structures and hence the stability of austenite and martensite phases. Therefore, knowledge of the effects of Cu addition is of great importance to understand the composition dependence of T_m and T_C. First-principles calculation results on Ni₈Mn_4-xGa₄Cu_x (x=0, 0.5, 1, 1.5 and 2) ferromagnetic shape memory alloys of this research draw following conclusions. The added Cu atom preferentially occupies the Mn site. The formation energy results indicate that ferromagnetic austenite is more stable than the paramagnetic one. The ferromagnetic state becomes instable and paramagnetic state becomes more stable when Mn is gradual substituted by Cu. The evaluated T_C decreases with increasing Cu content that is derived from the decrease of total energy difference between the paramagnetic and the ferromagnetic austenite. The experimentally observed decrease of T_m is originated from the decrease of total energy difference between the austenite and the non-modulated martensite. The difference between the up and down DOS is reduced with the increasing Cu content that gives rise to the decrease of the total magnetic moments. The purpose of this work is to explore the influence of Cu addition on crystal structure, T_m, T_C and electronic structures of Ni₈Mn_4-xGa₄Cu_x (x=0, 0.5, 1, 1.5 and 2) alloys by first-principles calculations, aiming at providing theoretical data and directions for developing high performance FSMAs.