  We perform a cosmological test of Cotton gravity, which describes gravity by
cotton tensor. The model we consider allows for the same background evolution
as the $\Lambda$CDM model. We derive the cosmological perturbation theory of
the scalar mode at the linear level, where the difference from the $\Lambda$CDM
model is characterized by the parameter $\beta$. We incorporate Cotton gravity
with a neutrino model and perform a Monte Carlo Markov Chain (MCMC) analysis
using data from the Cosmic Microwave Background (CMB) and Sloan Digital Sky
Survey (SDSS). The analysis constrains parameter
$\beta=-0.00008^{+0.00080}_{-0.00104}$ at the 1-$\sigma$ confidence level. We
conclude that currently, there is no obvious deviation between Cotton gravity
and the $\Lambda$CDM model in the linear cosmological perturbation level for
observations.
