In the planned linear accelerator called Compact Linear Collider, CLIC, electrons and positrons will be accelerated to velocities near the speed of light. A limiting factor in accelerating structures are vacuum breakdowns, which are electrical discharges from a surface as a result of a large electric field being applied. In the preparatory studies for the CLIC, Uppsala University in collaboration with The European Organization for Nuclear Research, CERN, is building a DC Spark system to analyze vacuum breakdowns. This system containing large planar electrodes will be cooled down all the way down to around 4 K in order to limit the rate of wich vacuum breakdowns happen. When cooling a system like this, which consists of different components made of different materials there is the question of how the system will be affected. The objective of this project is to investigate how the cooling will affect the stability in terms of stresses and to analyze the cool down time of the system. Another goal is to make a material recommendation for a few parts based on the results. This will be done by simulating the cooling in COMSOL Multiphysics, which is a program that uses finite element analysis to solve complex problems where different branches of physics interact. The conclusion is that the system will most likely be stable as it is and there is no need to redesign it. The choice of recommended material is alumina with the reason being it should cause the least stress and the smallest gap between the electrodes when the cooling is done. There was no big difference in the cool down time between the materials. Further studies and simulations on the system is also recommended since there are many factors not taken into consideration in this study.