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To the best of our knowledge, for the first time, the quantum Hall effect (QHE) is considered in thermal non-equilibrium conditions (TNEC) via [Formula: see text]-algebra. The [Formula: see text]-algebra is a topological tool that provides a controlling [Formula: see text]-parameter to approximate theoretical results to the laboratory ones. We obtain the QHE spectrum and [Formula: see text]-deformed Hall potential (or Hall potential in TNEC) via the density probability current. Because of the Pauli Exclusion Principle, we model the QHE using relativistic Fermi electron gas and the [Formula: see text]-deformed Fermi–Dirac distribution. We find the dependence of deformed Hall potential to temperature and magnetic field for different geometric [Formula: see text]-regions. Further, we geometrically prove that the magnitude of the Hall potentials depends on the [Formula: see text]-regions (i.e. [Formula: see text]). We find that for some [Formula: see text]-amounts, the Landau energy gap on graphene-based materials can be explained in the room-temperature quantum Hall effect (RTQHE). Our method may be applied to analyze new electronic devices based on graphene in TNEQ.

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