The law is extremely clear when it comes to governmental entities and their
meetings. It's called the "Open Meetings Law". It is found in the Section
551 of the Texas Government Code. Meetings must be open and the only matters
that can be discussed in private are those that fall under specific
exceptions which they can discuss privately in Executive Session, but they
must come back into the regular meeting and state on the record any specific
action that they took in Executive Session. If they took no action, they
also state that.

Here are some relevant sections regarding penalties:

SUBCHAPTER G. ENFORCEMENT AND REMEDIES; CRIMINAL VIOLATIONS
§ 551.141. Action Voidable
An action taken by a governmental body in violation of this chapter is
voidable.
Added by Acts 1993, 73rd Leg., ch. 268, § 1, eff. Sept. 1, 1993.

§ 551.142. Mandamus; Injunction
(a) An interested person, including a member of the news media, may bring an
action by mandamus or injunction to stop, prevent, or reverse a violation or
threatened violation of this chapter by members of a governmental body.
(b) The court may assess costs of litigation and reasonable attorney fees
incurred by a plaintiff or defendant who substantially prevails in an action
under Subsection (a). In exercising its discretion, the court shall consider
whether the action was brought in good faith and whether the conduct of the
governmental body had a reasonable basis in law.
Added by Acts 1993, 73rd Leg., ch. 268, § 1, eff. Sept. 1, 1993.

§ 551.143. Conspiracy to Circumvent Chapter; Offense; Penalty
(a) A member or group of members of a governmental body commits an offense if
the member or group of members knowingly conspires to circumvent this chapter
by meeting in numbers less than a quorum for the purpose of secret
deliberations in violation of this chapter.
(b) An offense under Subsection (a) is a misdemeanor punishable by:
(1) a fine of not less than $100 or more than $500;
(2) confinement in the county jail for not less than one month or more than
six months; or
(3) both the fine and confinement.
Added by Acts 1993, 73rd Leg., ch. 268, § 1, eff. Sept. 1, 1993.

§ 551.144. Closed Meeting; Offense; Penalty
(a) A member of a governmental body commits an offense if a closed meeting is
not permitted under this chapter and the member knowingly:
(1) calls or aids in calling or organizing the closed meeting, whether it is
a special or called closed meeting;
(2) closes or aids in closing the meeting to the public, if it is a regular
meeting; or
(3) participates in the closed meeting, whether it is a regular, special, or
called meeting.
(b) An offense under Subsection (a) is a misdemeanor punishable by:
(1) a fine of not less than $100 or more than $500;
(2) confinement in the county jail for not less than one month or more than
six months; or
(3) both the fine and confinement.
(c) It is an affirmative defense to prosecution under Subsection (a) that the
member of the governmental body acted in reasonable reliance on a court order
or a written interpretation of this chapter contained in an opinion of a
court of record, the attorney general, or the attorney for the governmental
body.
Added by Acts 1993, 73rd Leg., ch. 268, § 1, eff. Sept. 1, 1993.
Amended by Acts 1999, 76th Leg., ch. 647, § 3, eff. Aug. 30, 1999.

§ 551.145. Closed Meeting Without Certified Agenda or Tape Recording;
Offense; Penalty
(a) A member of a governmental body commits an offense if the member
participates in a closed meeting of the governmental body knowing that a
certified agenda of the closed meeting is not being kept or that a tape
recording of the closed meeting is not being made.
(b) An offense under Subsection (a) is a Class C misdemeanor.
Added by Acts 1993, 73rd Leg., ch. 268, § 1, eff. Sept. 1, 1993.

§ 551.146. Disclosure of Certified Agenda or Tape Recording of Closed
Meeting; Offense; Penalty; Civil Liability
(a) An individual, corporation, or partnership that without lawful authority
knowingly discloses to a member of the public the certified agenda or tape
recording of a meeting that was lawfully closed to the public under this
chapter:
(1) commits an offense; and
(2) is liable to a person injured or damaged by the disclosure for:
(A) actual damages, including damages for personal injury or damage, lost
wages, defamation, or mental or other emotional distress;
(B) reasonable attorney fees and court costs; and
(C) at the discretion of the trier of fact, exemplary damages.
(b) An offense under Subsection (a)(1) is a Class B misdemeanor.
(c) It is a defense to prosecution under Subsection (a)(1) and an affirmative
defense to a civil action under Subsection (a)(2) that:
(1) the defendant had good reason to believe the disclosure was lawful; or
(2) the disclosure was the result of a mistake of fact concerning the nature
or content of the certified agenda or tape recording.
Added by Acts 1993, 73rd Leg., ch. 268, § 1, eff. Sept. 1, 1993.