, 2010

Let $p$ be any prime number. If $p$ is not divisible by $7$, then $p^3$ is congruent to either $1$ or $-1$ modulo $7$. Since $2008 \equiv -1 \pmod{7}$ and $2010 \equiv 1 \pmod{7}$, either of the numbers $p^3 + 2008$ and $p^3 + 2010$ is divisible by $7$ and hence composite. If $p$ is divisible by $7$, then $p = 7$ and $p^3 + 2010 = 7^3 + 2010 = 2353 = 13 \cdot 181$ is composite, too.