We prove that in rotationally invariant first-passage percolation on $\mathbb{R}^d$ with $d\geq2$, the passage time $T_n$ over a distance $n$ satisfies $\mathbb{E}[\exp(|T_n-μn|/n^{\frac{1}{2}(1-\varepsilon)})]\leq C$ for some $\varepsilon>0$, where $μ$ is the time constant. This implies that its variance is at most a constant times $n^{1-\varepsilon}$, the bound that Basu, Sidoravicius and Sly proved in the planar case. As in their work, our argument is an induction on the scale, but in place of their proof that the geodesic is chaotic, we use the averaging argument of Benjamini, Kalai and Schramm together with a logarithmic improvement of the Efron-Stein inequality due to Benaïm and Rossignol.
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