  This paper is concerned about the lifespan estimate to the Cauchy problem of
semilinear damped wave equations with the Fujita critical exponent in high
dimensions$(n\geq 4)$. We establish the sharp upper bound of the lifespan in
the following form \begin{equation}\nonumber\\ \begin{aligned}
T(\varepsilon)\leq \exp(C\varepsilon^{-\frac 2n}), \end{aligned} \end{equation}
by using the heat kernel as the test function.
