Context. Diverse studies have shown that it is important to consider the impact of metallicity on the chemodynamical evolution of protoplanetary disks. It has been suggested that there may be different chemistry cycles in non-solar metallicity environments at work or that the efficiency of mass transport in protostars and pre-main-sequence stars is dependent on metallicity to a certain extent. Aims. We study the influence of different metallicities on the physical, thermal, and chemical properties of protoplanetary disks, particularly with regard to the formation and destruction of carbon-based molecules. Methods. With the thermo-chemical code ProDiMo (PROtoplanetary DIsk MOdel), we investigated the impact of lower metallicities on the radiation field, disk temperature, and the abundance of different molecules (H2O, CH4, CO, CO2, HCN, CN, HCO+, and N2H+). We used a fiducial disk model as a reference and produced two derivative models based on lower metallicity. We studied the resulting influence on different chemical species by analyzing their abundance distribution throughout the disk and their vertical column density. Furthermore, we examined the formation and destruction reactions of the chemical species. Results. Our results demonstrate a relation between the metallicity of the disk and the strength of the stellar radiation field inside the disk. As the metallicity decreases, the radiation field is capable of penetrating deeper regions of the disk. As a result, there is a stronger radiation field in the disk overall with lower metallicity, which also heats up the disk. This triggers a series of changes in the chemical formation and destruction efficiencies for different chemical species. In most cases, the available species abundances change and have greater values compared to scaled-down abundances by constant factors. Metallicity has a clear impact on the snowline of the molecules studied here as well. As metallicity decreases the snowlines are pushed further out and existing snow rings shrink in size. Conclusions. We find that the abundances of the studied molecules in lower metallicity disks cannot be understood or reproduced by scaling down the respective species abundances of the reference disk model. This is because the chemical reactions responsible for the destruction and formation of the studied molecules change as the metallicity of the disk is reduced. We found a strong overabundance (relative to scaled-down values) in the models with lower metallicity for gaseous species (CN, CO, HCO+, N2H+), which are particularly useful in observations. This could be advantageous for future observations in low-metallicity environments. Further studies considering different aspects of the disk are needed to gain a deeper understanding of the relation between metallicity and disk thermochemical evolution. Future studies ought to consider other processes, such as different dust grain size distribution, different stellar radiation fields, and stellar burst scenarios.