The formula  you mentioned is true for a smooth curve $C$ on a unit sphere of any dimension and it is a special case  of the Kac-Rice formula; see [Example 15 here][1].  

Let me explain a bit this point of view. Suppose $C$ is a   smooth curve  on the unit sphere $S^n$ in $\newcommand{\bR}{\mathbb{R}}$ $\bR^{n+1}$. Fix an arclength parametrization of $C$

$$ [0,L]\ni s \mapsto \big( x_0(x),\dotsc, x_n(s)\big)\in \bR^{n+1}, $$

where $L$ is the length of the curve $(x_0,\dotsc,x_n)$ are the canonical coordinates in $\bR^{n+1}$ and

$$
\sum_{k=0}^n x_k(s)^2=1,\;\;\forall s. 
$$
 Fix  independent standard normal random variables $U_0,U_1,\dotsc, U_n$ and form the random function
$$
F:=[0,L]\to\bR,\;\;F(s)=\sum_{k=0}^m U_k x_k(s).
$$
The zeros of $F$ correspond  to the intersections of $C$ with the Equator
$$ \sum_{k=0}^n U_k x_k=1,\;\;\sum_{k=0}^nx_k^2=1.$$

Denote by $N$ the number of zeros of $F$.  Another version  of the Kac-Rice formula will give you a description  of the second combinatorial moment $\bE[N(N-1)]$ see Theorem 3.2  in the book  *Level Sets and  Extrema of Random Fields and Processes*, by J.M. Azaïs and M. Wschebor, John Wiley & Sons, 2009.

It involves certain conditional expectations. If $L<2\pi$ these conditional expectations are  more manageable.  If $L>2\pi$ it is conceivable that the in some cases the variance is infinite. This is  a speculation not a hard fact.


  [1]: https://www3.nd.edu/~lnicolae/Kac_Rice.pdf