Given that "the quantum zeno effect can "freeze" the evolution of the system by measuring it frequently enough in its known initial state." Is there any equivalent/similar Zeno effect in quantum gravity (theories)?
Quantum mechanics itself is separate from any particular set of physical laws. The physical laws include things like electrons and photons and gravitons and their interactions. Quantum mechanics itself includes things like the uncertainty principle, the Schrödinger equation, Bell's theorem, and qubits. These are all in some sense "prior" to any particular physical theory, at a level of reality that we never imagined might need revision before quantum mechanics was discovered. Scott Aaronson described quantum mechanics as "the operating system that other physical theories run on", which is an analogy that may or may not be helpful.
The quantum Zeno effect belongs to quantum mechanics itself. As such, it should work in quantum gravity just as it does in nongravitational physics. If it doesn't, then gravity is not really quantum, in the sense that we currently use that term.
Whether we could experimentally detect the effect is another matter. Right now, we can't detect any quantum gravitational effect. The Large Hadron Collider has a collision energy of 13 TeV, while quantum gravity is expected to become relevant at a collision energy of around 1,000,000,000,000,000 TeV (the Planck energy). There was some hope that (because of large extra dimensions) this value was incorrect, and the real energy scale of quantum gravity was so much lower that it might even be seen at the LHC. But that hasn't panned out so far.