Many articles have been published on the uncertainty analysis of high temperature gas kinetic systems that are based on detailed reaction mechanisms. In all these articles a temperature independent relative uncertainty of the rate coefficient is assumed, although the chemical kinetics databases suggest temperature dependent uncertainty factors for most of the reactions. The temperature dependence of the rate coefficient is usually parameterized by the Arrhenius equation. An analytical expression is derived that describes the temperature dependence of the uncertainty of the rate coefficient as a function of the elements of the covariance matrix of the Arrhenius parameters. Utilization of the joint uncertainty of the Arrhenius parameters is needed for a correct uncertainty analysis in varying temperature chemical kinetic systems. The covariance matrix of the Arrhenius parameters, the lower and upper bounds for the rate coefficient, and the temperature interval of validity together define a truncated multivariate normal distribution of the transformed Arrhenius parameters. Determination of the covariance matrix and the joint probability density function of the Arrhenius parameters is demonstrated on the examples of two gas-phase elementary reactions.