Should I use the discounted average reward as objective in a finite-horizon problem?

I am new to reinforcement learning, but, for a finite horizon application problem, I am considering using the average reward instead of the sum of rewards as the objective. Specifically, there are a total of $T$ maximally possible time steps (e.g., the usage rate of an app in each time-step), in each time-step, the reward may be 0 or 1. The goal is to maximize the daily average usage rate.

Episode length ($T$) is maximally 10. $T$ is the maximum time window the product can observe about a user’s behavior of the chosen data. There is an indicator value in the data indicating whether an episode terminates. From the data, it is offline learning, so in each episode, $T$ is given in the data. As long as an episode doesn’t terminate, there is a reward of ${0, 1}$ in each time-step.

I heard if I use an average reward for the finite horizon, the optimal policy is no longer a stationary policy, and optimal $Q$ function depends on time. I am wondering why this is the case.

I see normally, the objective is defined maximizing

$$sum_t^T gamma^t r_t$$

And I am considering two types of average reward definition.

1. $1/T(sum^?_{t=0}gamma^t r_t)$, $T$ varies is in each episode.

2. $1/(T-t)sum^T_{i=t-1}gamma^i r_i$