Hi,
I was staying in Texas for period of 6 months and had taken an apartment on rent
I had given a notice period of 2 months in advance before I was vacating the apartment as per the rules
However at the end of 1st month of notice period, I had to urgently go out of Texas and I subleased the apartment for the remaining 2nd month to my friend and signed the appropriate document
Now I have come to know from the leasing agent/company, then they have got my signature on one of documents which lists that I had requested to cancel my notice period a month before I was about to vacate and when I subleased the apartment to my friend and they are asking me to pay the lease breakage charges for it, otherwise they will proceed with legal action against me
***. Guide me on what does the US law state about it and how I can resolve this issue

First ask to see the document that rescinded your vacate order. Then review your lease to ensure that sub-letting was allowed.

Neither Texas nor US law really applies here. The key issues is what you did with respect to the terms of the lease.

You have to comply with the term of the lease you signed with the Owner ( or leasing agency in this case).

If you dated the notice incorrectly a month too early, that falls on you, not them. And they would be owed that money... and any sublet agreement you had will be voided, even when a sublet is allowed they are always subordinate to the terms and conditions of the primarily lease.