Low-temperature specific heat and thermal conductivity measurements have been performed on cesium borate glasses as a function of cesium oxide content. We have found experimental evidence of a concurrent growth of specific heat and thermal conductivity with increasing Cs+ content. This finding shows the existence of an uncommon relationship between the peak in C-p/T-3 and the plateau in thermal conductivity in glasses and represents the most intriguing result for these alkaline borate glasses. The role of local modes associated with heavy cations on the vibrational dynamics in oxide glasses has been considered. Furthermore, a possible correlation between low-temperature thermal properties and the structure on the nanometer length scale of these glasses is put forward.