The way to study the topology of the situation was introduced by Khovanski in "Newton polyhedra, and toroidal varieties" Funkcional. Anal. i Priložen. 11 (1977), no. 4, 56--64, 96. His result (if I have interpreted it correctly) is that $X$ may be compactified as a hypersurface in a projective toric variety to a smooth variety with normal crossings such that each stratum is of the same form as $X$. As far as I can see this construction works uniformly so that we would get the same construction over a (suitable) mixed characteristic discrete valuation ring. Then desired isomorphism then follows from the smooth and proper base change theorem. (I have some vague recollection that this comment is also to be found somewhere in SGA but I am not going to do any wading looking for it...)
Addendum: Let me first note that the right setup to even formulate the question is a scheme $S$ with functions on it giving the coefficients of $f$. The latter polynomial should then be non-degenerate in the sense that all its fibres over (geometric) points of $S$ should be non-degenerate. The statement is then that if $\pi\colon X\to S$ is the scheme of zeroes of $f$ in the constant torus over $S$, then $R^i\pi_\ast\mathbb Z_\ell$ is locally constant commuting with base change where $\ell$ is invertible in $\mathcal O_S$ (together with comparison theorem of $\ell$-adic cohomology and classical for a complex point of $S$). As this statement is only dealing with the $\ell$-adic sheaves $R^i\pi_\ast\mathbb Z_{\ell}$ we may use the definition of $\ell$-adic sheaf introduced by Jouanolou in SGA V. Kohvanski's method then should give a compactification of $X$ by a smooth $S$-scheme with complement of relative normal crossings. The two theorems used, proper base change and vanishing of vanishing cycles, then follows directly from the case of finite coefficients (by Jouanolou's very definition).
The way to study the topology of the situation was introduced by Khovanski in "Newton polyhedra, and toroidal varieties" Funkcional. Anal. i Priložen. 11 (1977), no. 4, 56--64, 96. His result (if I have interpreted it correctly) is that $X$ may be compactified as a hypersurface in a projective toric variety to a smooth variety with normal crossings such that each stratum is of the same form as $X$. As far as I can see this construction works uniformly so that we would get the same construction over a (suitable) mixed characteristic discrete valuation ring. Then desired isomorphism then follows from the smooth and proper base change theorem. (I have some vague recollection that this comment is also to be found somewhere in SGA but I am not going to do any wading looking for it...)