Second order weakly stationary process I was wondering, whether a stochastic process with a covariance, that is not depending on time, imply constant variances. Since I read a textbook, where they call a process, which is second order weakly stationary also covariance stationary.

Actually, it does. If you have a stochastic process $(X_t)$, with $Cov(X_t,X_s)$ that does not depend on time, then
$$
V(X_t)=Cov(X_t,X_t)
$$
does not depend on $t$ either. Of course, the question remains whether or not $V(X_t)$ is finite. Usually, some parameter restrictions need to be satisfied to guarantee that.
For example, an AR(1) process $X_t=\phi_0+\phi_1X_{t-1}+\epsilon_t$, $\epsilon \sim \mathcal N(0,\sigma^2)$ has autocorrelation function $\gamma_k=\phi_1\gamma_{k-1}$ that does not depend on $t$ and the variance is given by $\gamma_0=\frac{\sigma^2}{1-\phi_1^2}$ which is also constant and finite as long as $\vert \phi_1 \vert <1$.
