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We prove a general lower bound to the spectral gap of the Davies generatorfor Hamiltonians that can be written as the sum of commuting Pauli operators.These Hamiltonians, defined on the Hilbert space of $N$-qubits, serve as one ofthe most frequently considered candidates for a self-correcting quantum memory.A spectral gap bound on the Davies generator establishes an upper limit on thelife time of such a quantum memory and can be used to estimate the time untilthe system relaxes to thermal equilibrium when brought into contact with athermal heat bath. The bound can be shown to behave as $\lambda \geq {\calO}(N^{-1}\exp(-2\beta \, \overline{\epsilon}))$, where $\overline{\epsilon}$ isa generalization of the well known energy barrier for logical operators.Particularly in the low temperature regime we expect this bound to provide thecorrect asymptotic scaling of the gap with the system size up to a factor of$N^{-1}$. Furthermore, we discuss conditions and provide scenarios where thisfactor can be removed and a constant lower bound can be proven

Thermalization time bounds for Pauli stabilizer Hamiltonians