Next generation use-cases of wireless IP networks, including especially the Massive Machine Type Communications and IoT applications for the distribution of sensory and similar data will require the capability to handle large number of connections while maintaining a low-power footprint in order to function efficiently during long-term deployments. The reduction of the packetisation overhead resulting from the adaptation of header compression could potentially decrease the battery usage of network heavy applications via diminished wireless interface activity. Normally header compression is employed to minimise the overhead of IP-based cellular traffic between two connected peers. This paper presents for the first time comparative power consumption measurements for Robust Header Compression version 2 (RoHCv2) using WiFi and LTE. We find that the adoption of header compression on modern mobile devices will generally not result in increased power consumption based on the extra complexity added by the execution of the algorithms. We also show that the usage of RoHCv2 can potentially even decrease the battery drain on average by about 0.05 W when payloads are small.