FLRW in $n+1$ dimensions

Can you construct, using Newtoian gravity or the Riemann-tensor for a maximally symmetric metric, a FLRW-universe in \(n+1\) dimensions? Are the Friedmann-equations different?

bonus question: Please show that the equation of state of photons in $n+1$ have an equation of state of \(w=1/n\) and that they dilute propto $a^{-(n+1)}$ under the Hubble-expansion?