The entropic quasi-de Sitter instability time from the distance conjecture

From the entropy argument for the dS swampland conjecture which connects the Gibbons-Hawking entropy bound with the distance conjecture, we find the entropic quasi-dS instability time given by $1/(\sqrt{\epsilon_H}H)\log(m_{\rm Pl}/H)$ as the lifetime of quasi-dS spacetime. It depends on the slow-roll parameter, and contains the logarithmic factor $\log (m_{\mathrm{P}\mathrm{l}}\mathrm{/}H)$ which can be found in the scrambling (or decoherence) time as well. Such a logarithmic factor enhances the geodesic distance of the modulus from the mere Planck scale, and also possibly relaxes the bound on $m_{\mathrm{P}\mathrm{l}}^2{\mathrm{\nabla }}^{2}V\mathrm{/}V$.