The $W$ boson mass $m_W$ in the grand unified theory inspired $SO\left(5\right)\times U\left(1\right)\times SU\left(3\right)$ gauge-Higgs unification in the Randall-Sundrum (RS) warped space is evaluated. The muon decay ${\mu }^{-}\to e^{-}{\overline{\nu }}_e{\nu }_{\mu }$ proceeds by the exchange of not only the zero mode of the $W$ boson $\left(W^{\left(0\right)}\right)$ but also Kaluza-Klein (KK) excited modes $W^{\left(n\right)}$ and $W_R^{\left(n\right)}$ ($n\ge 1$) at the tree level. The anti–de Sitter curvature of the RS space also affects the relationship among the gauge couplings and the ratio of $m_W$ to the $Z$ boson mass $m_Z$. The $W$ couplings of leptons and quarks also change. With the given KK mass scale $m_{\mathrm{KK}}$ the range of the Aharonov-Bohm phase ${\theta }_H$ in the fifth dimension is constrained. For $m_{\mathrm{KK}}=13\mathrm{TeV}$, $0.085\lesssim {\theta }_H\lesssim 0.11$ and $80.381\lesssim m_W\lesssim 80.407\mathrm{GeV}$. The predicted value of $m_W$ for $13\le m_{\mathrm{KK}}\le 20\mathrm{TeV}$ lies between $m_W^{\mathrm{SM}}=80.354\pm 0.007\mathrm{GeV}$ in the standard model and $m_W^{\mathrm{CDF}}=80.4335\pm 0.0094\mathrm{GeV}$, the value reported by the CDF Collaboration in 2022.