I am unfamiliar with the specifics of T5, but in general, LLMs cannot easily learn "new" information after they have been trained/initialized with sample data. Adding new sample data could require a large degree of processing, possibly equal to the original amount of work to train the model.
Instead, new information is typically injected via each query's input (e.g. prompt). Various models have different input size limits (usually represented as tokens). Inquiries can be made about the new data by placing it alongside the query. For instance, if an LLM client has access to the web, it could convert a webpage into text, then send the text along with a user's query to the LLM. However, it would have to do this for each query or session. Once a session is complete (or the input size is reached), the LLM will either start "forgetting" the new information or will not allow any additional queries.
Some LLMs (like ChatGPT) have plugins available for automatically injecting an information source into each prompt or conversation, making the process more transparent to the user. But these still consume input/token space and the amount of information that can be given to the LLM is limited.