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Let $A$ be a CM-finite Artin algebra with a Gorenstein-Auslander generator $E$, $M$ be a Gorenstein projective $A$-module and $B={\mathrm{End}}_{A}M$. We give an upper bound for the finitistic dimension of $B$ in terms of homological data of $M$. Furthermore, if $A$ is $n$-Gorenstein for $2\le n<\infty$, then we show the global dimension of $B$ is less than or equal to $n$ plus the $B$-projective dimension of ${\mathrm{Hom}}_A(M,E).$ As an application, the global dimension of ${\mathrm{End}}_{A}E$ is less than or equal to $n$.