Train LSTM model with multiple time series
As you mention, all rooms have similar properties, thus it makes sense developing a single LSTM model for all of them.

You have to pre-process your data so that you have input vectors $X_i$ of 7 time steps and 1 time step as label vector $Y_i$:

$
X_i = [x_i[t-6 ],x_i[t-5 ],x_i[t-4 ],x_i[t-3 ],x_i[t-2],x_i[t-1 ],x_i[t]]\\
Y_i = [x_i[t+1]]
$

where the subscript $i$ denotes the $i_{th}$ vector.

Let's imagine you can generate 100 vectors for each room, you'll end up with a dataset of $10^4$ input vectors. The number of vector per room will depend on the overlap you use, for instance, if you have data for 100 days per each room and you use an overlap of 99 days (your window slides one day in the future), you'll have 99 vectors for each room.

To train the LSTM you use the typical Mini-batch training. Make sure you don't propagate the state for batch sample $i$ to sample $i+1$ in order to treat them individually (in Keras you set the stateful flag to False)