Let's say I choose threshold in logistic regression equals to $0.8$. If it is lower then class is $0$ else is $1$. Then, how do I interpret the outcome on test point $x$, $h(x)=0.6$, where $h$ is my logistic regression model. I think I cannot say that with probability equals $0.6$ class of $x$ is $1$? Is there any way to interpret this? Should I just scale appropriate intervals so in that case the outcome $0.6$ I can interpret as probability of $0.375$ that point is in class $1$?
You have to be careful to distinguish the probability estimate from a logistic regression model from the probability cutoff you use when you apply the model in practice to make a decision.
Logistic regression results can be expressed in terms of the probability of class membership. I take your terminology $h(x) = 0.6$ to mean that your model $h$ predicts that case $x$ has probability of 0.6 of belonging to Class 1. That's the probability estimate from the model. It has nothing to do directly with a probability cutoff.
When you apply the probability model, practical considerations come into play. Say that false-positive decisions about membership in Class 1 cost more than do false-negative decisions. Then you would want to avoid a decision that places a true Class 0 case into Class 1. Using a high probability cutoff like 0.8 to make assignments of class membership for your purpose would accomplish that.
So if a case has $h(x) = 0.6$ that still means it has a 60% chance of actually being in Class 1. You just will have made a decision not to assign it to Class 1 for your purposes because you don't want to risk that it really is in Class 0.