I found and answer to my own question. So, I post it here for people who maybe have the same problem. The answer, however, is quite intuitive. The last observation used for the estimation of the physical density is also the time point where the investors know the most about the physical density because at this point the most possible historical observations are used. Hence, they evaluate the future rationally based on this pseudo-true measure. Therefore, the option prices at the last observed time point need to be used to estimate the risk neutral density.

So, the estimation procedure for a empirical pricing kernel could be as follows:

1. Get the closing prices of the considered index in a reasonable time interval $[t,T]$
2. Get the option prices for the index at time $T$ with equal time to expiration but different strike prices
3. Estimate the physical density $\hat p$ with data obtained in the first step
4. Estimate the risk neutral density $\hat q$ with date obtained in the second step
5. Calculate the empirical pricing kernel as $\hat k = \frac{\hat p}{\hat q}$

Note: The empirical pricing kernel equal is also referred to as the stochastic discount factor.

I found and answer to my own question. So, I post it here for people who maybe have the same problem. The answer, however, is quite intuitive. The last observation used for the estimation of the physical density is also the time point where the investors know the most about the physical density because at this point the most possible historical observations are used. Hence, they evaluate the future rationally based on this pseudo-true measure. Therefore, the option prices at the last observed time point need to be used to estimate the risk neutral density.

So, the estimation procedure for a empirical pricing kernel could be as follows:

1. Get the closing prices of the considered index in a reasonable time interval $[t,T]$
2. Get the option prices for the index at time $T$ with equal time to expiration but different strike prices
3. Estimate the physical density $\hat p$ with data obtained in the first step
4. Estimate the risk neutral density $\hat q$ with date obtained in the second step
5. Calculate the empirical pricing kernel as $\hat k = \frac{\hat q}{\hat p}$

Note: The empirical pricing kernel equal is also referred to as the stochastic discount factor.