Fusion power offers the prospect of a safe and clean sustainable energy source, and is of increasing importance for meeting the world energy demand and curbing CO2 emissions. For an indirect-driven inertial confinement cryogenic target, the D-T ice layer inside the capsule should have a uniformity more than 99% and an inner surface roughness less than a root mean square value of 1 m to avoid Rayleigh-Taylor instabilities. And this highly smooth ice layer required for ignition is considered to be affected by the thermal environment around the fuel capsule. In the present study, a numerical investigation is conducted to examine the static and dynamic characteristics of the thermal environment outside the fuel capsule. Numerical model is proposed and verified by a simplified cryogenic target, and the calculated temperature distribution around the capsule shows to be in good agreement with the experimental data. Based on the established model, the propagation of periodic disturbance of cooling wall temperature in the hohlraum is investigated, and the relations between the temperature disturbance on the cooling wall and the temperature distribution around the capsule surface are obtained. The effects of disturbance amplitude, the disturbance period, and the hohlraum gas composition on the propagation process are investigated separately. The results indicate that for stable cooling temperature, the thermal environment around the capsule shows certain dependence on the gas filled in the hohlraum. The temperature uniformity of the capsule outer surface deteriorates with the increase of fill gas pressure but can be improved by increasing the He content of the filling gas mixture. At an oscillating cooling temperature, the attenuation of amplitude is significant when the periodic disturbance propagates from the cooling rings to the hohlraum and to the capsule surface. For the sine wave form disturbance investigated in the present study, shorter disturbance period results in larger attenuation of the disturbance amplitude. Higher gas pressure leads to smaller amplitude of average temperature on the capsule outer surface. The propagation process of cooling temperature disturbance also demonstrates dependence on the filling gas composition. The higher fraction of H2 in the He-H2 mixture helps to attenuate the disturbance amplitude and suppress the propagation of the temperature disturbance. However, the temperature uniformity around the capsule exhibits different characteristics from cooling temperature disturbance. Under the oscillating cooling conditions, moderate period, lower amplitude, lower pressure and higher fraction of He in the He-H2 mixture help to improve the temperature uniformity around the capsule. The results are of guiding significance for determining the controlling scheme in experiment and further design option for the cryogenic target.