This is a stochastic process problem on a book:
A stochastic process $X(t) \equiv Y$ where $Y$ is a random variable and $E(Y)=a, D(Y) = \sigma^2$. Calculate the autocovariance of $X(t)$.
The answer on the book is $\sigma^2$, but I think that this is only for $C_X(t,t)$ case. For $C_X(s,t)$, since we don't know whether $X(s)$ and $X(t)$ are related, the answer is unknown. Am I correct?
The answer given in the book is correct.
Notice that the definition of $X(t)$ doesn't just say that $X(t)$ is distributed as $Y$ for every $t \geq 0$ (or some similar condition). It says that $X(t) = Y$ for every $t \geq 0$. This means that for any $s,t$ we have that $X(t) = X(s)$ (i.e. $X$ is a constant process).