Are multi-qubit control RX gates scaling exponentially?

An $n$-qubit controlled phase gate with error $epsilon$ takes $O(n + lg frac{1}{epsilon})$ gates to achieve.

The $O(n)$ dependence is easiest to understand in the case where you have $n$ ancillae:

The $O(lg frac{1}{epsilon})$ dependence is from the need to decompose the single qubit phase rotation into the gateset that is actually supported, e.g. using repeat-until-success circuits.

Only a single ancilla is actually required. And if you're willing to increase the cost to $O(n cdot lg frac{1}{epsilon})$ and use $n$ arbitrary single qubit phase rotations instead of one then no ancillae are needed at all.