Ozone is an important radiative trace gas in the Earth's atmosphere. The presence of ozone can significantly influence the thermal structure of an atmosphere, and by this e.g. cloud formation. Photochemical studies suggest that ozone can form in carbon dioxide-rich atmospheres. We investigate the effect of ozone on the temperature structure of simulated early Martian atmospheres. With a 1D radiative-convective model, we calculate temperature-pressure profiles for a 1 bar carbon dioxide atmosphere. Ozone profiles are fixed, parameterized profiles. We vary the location of the ozone layer maximum and the concentration at this maximum. The maximum is placed at different pressure levels in the upper and middle atmosphere (1-10 mbar). Results suggest that the impact of ozone on surface temperatures is relatively small. However, the planetary albedo significantly decreases at large ozone concentrations. Throughout the middle and upper atmospheres, temperatures increase upon introducing ozone due to strong UV absorption. This heating of the middle atmosphere strongly reduces the zone of carbon dioxide condensation, hence the potential formation of carbon dioxide clouds. For high ozone concentrations, the formation of carbon dioxide clouds is inhibited in the entire atmosphere. In addition, due to the heating of the middle atmosphere, the cold trap is located at increasingly higher pressures when increasing ozone. This leads to wetter stratospheres hence might increase water loss rates on early Mars. However, increased stratospheric H2O would lead to more HOx, which could efficiently destroy ozone. This result emphasizes the need for consistent climate-chemistry calculations to assess the feedback between temperature structure, water content and ozone chemistry. Furthermore, convection is inhibited at high ozone amounts, leading to a stably stratified atmosphere.