Fig. presents the temperature dependence of the VSC rate constant, plotting as ln(k/k 0 ) as a function of 1/T . The cavity lifetime is fixed as τ c = 1000 fs, and the cavity frequency is ω c = ω 0 . Fig. shows the Eyring-type plots for reactions outside the cavity (black points) and inside a resonant cavity under various light-matter coupling strengths. The rate constants were obtained from HEOM simulations (dots), and fitted by the least square to obtain linearity (thin lines). One can see that as Ω R increases, the slope of the Eyring plots becomes more negative (an increasing the effective activation enthalpy). Meanwhile, the effective entropy also increased significantly as one increase Ω R . Fig. presents the change of the effective activation enthalpy ∆∆H ‡ as increasing Ω R . The HEOM results for ∆∆H ‡ (blue open circles with thin guiding lines) are extracted from the slopes of the fitted lines in Fig. . Further, the FGR results (gold dashed line) are presented, in which k VSC is calculated using Eq. 17 (re-scale by a factor of 0.5) and plug-in Eq. 24a to obtain the cavity induced change ∆∆H ‡ . When k VSC is small, Eq. 24a is proportional to k VSC , i.e., proportional to Ω 2 R according to the analytic FGR rate theory (see Eq. 17). One can see that from Fig. when Ω R < 15 cm -1 , ∆H ‡ c increases quadratically with Ω R , and the FGR results agree with the HEOM results. When Ω R > 15 cm -1 , the behavior deviates from quadratic scaling, and FGR results still closely match the trend of HEOM results. Fig. presents the change of the effective activation entropy ∆∆S ‡ , with results obtained from HEOM (blue open circles with thin line) and FGR (golden dashed line), where the FGR also provides a good agreement with the exact results. Note that Eq. 24 also works well in the lossy regime, where the results with τ c = 100 fs and k/k 0 evaluated using Eq. 10 are presented in Fig. of Supplementary Note 7.