It is indeed true unless $X$ is a cubic plane curve or a quartic surface. The reason is the following: $\varphi ^*$ preserve the canonical class, which is $(d-n-1)H$ (for $X$ a smooth hypersurface of degree $d$ in $\mathbb{P}^n$). Since $H^2(X,\mathbb{Z})$ has no torsion, this implies $\varphi ^*H=H$ if $d\neq n+1$. Now if $n\geq 4$, we have $H^2(X,\mathbb{Z})=\mathbb{Z}$ by Lefschetz theorem, which implies again $\varphi ^*H=H$. The remaining cases are cubic plane curves and quartic surfaces, and indeed in those cases there are examples of $\varphi $ with $\varphi ^*H\neq H$.