Kobayashi et al.\ have recently shown that various verification problems for higher-order functional programs can naturally be reduced to the validity checking problem for $\text{HFL}\mathbb{Z}$, a higher-order fixpoint logic extended with integers. We propose a refinement type system for checking the validity of $\nu\text{HFL}\mathbb{Z}$ formulas, where $\nu\text{HFL}\mathbb{Z}$ is a fragment of $\text{HFL}\mathbb{Z}$ without least fixpoint operators, but sufficiently expressive for encoding safety property verification problems. Our type system has been inspired by the type system of Burn et al. for solving the satisfiability problem for HoCHC, which is essentially equivalent to the $\nu\text{HFL}\mathbb{Z}$ validity checking problem. Our type system is more expressive, however, due to a more sophisticated subtyping relation. We have implemented a type-based $\nu\text{HFL}\mathbb{Z}$ validity checker ${\rm R{\small e}THFL}$ based on the proposed type system, and confirmed through experiments that ${\rm R{\small e}THFL}$ can solve more instances than Horus, the tool based on Burn et al.’s type system.