In terms of OSI layer 7 protocol, Matrix uses HTTP, though in the future it may specify other application protocols.
The problem with HTTP is that it drains 11 times more battery on a mobile phone than something like MQTT.
Can you give any evidence to support that statement?
As Perceptes says, Matrix is designed to have modular transports. So if you want to use MQTT or CoAP or WebSockets or whatever for the client/server API, then go for it :) PRs welcome!