  We provide an example of a nonnegative $k$-regulous function on
$\mathbb{R}^n$ for $k\geq 1$ and $n \geq 2$ which cannot be written as a sum of
squares of $k$-regulous functions. We then obtain lower bounds for Pythagoras
numbers $p_{2d}(\mathcal{R}^k(\mathbb{R}^n))$ of $k$-regulous functions on
$\mathbb{R}^n$ for $k\geq 1$ and $n\geq 2$. We also prove that the second
Pythagoras number of the ring of $0$-regulous functions $\mathcal{R}^0(X)$ on
an irreducible $0$-regulous affine variety $X$ is finite and bounded from above
by $2^{\dim X}$.
