Based on the pseudospin model of the deformed CsH$_2$PO$_4$ crystal within the Glauber method, the equation for the time-dependent mean value of the pseudospin is obtained, which is solved in the case of small deviations from the equilibrium state. Using the solution of the equation, we find expressions for the longitudinal dynamic dielectric constant and relaxation time. Based on the proposed parameters of the theory, the temperature and frequency dependences of the dynamic dielectric constant and the temperature dependence of the relaxation time are calculated and investigated. A detailed numerical analysis of the obtained results was performed. The influence of hydrostatic pressure on the dynamic characteristics of CsH$_2$PO$_4$ is studied.