diff --git a/Library/Homebrew/livecheck/strategy.rb b/Library/Homebrew/livecheck/strategy.rb
index abe14e200f868..3e9e1bef7c032 100644
--- a/Library/Homebrew/livecheck/strategy.rb
+++ b/Library/Homebrew/livecheck/strategy.rb
@@ -65,13 +65,16 @@ module Strategy
 
       # Baseline `curl` options used in {Strategy} methods.
       DEFAULT_CURL_OPTIONS = {
-        print_stdout:    false,
-        print_stderr:    false,
+        # By default, `Utils::Curl` methods will print text (e.g. the command)
+        # when `--debug` is set. `debug` is set to `false` to ensure that this
+        # unwanted text doesn't appear in livecheck's debug output.
         debug:           false,
-        verbose:         false,
         timeout:         CURL_PROCESS_TIMEOUT,
         connect_timeout: CURL_CONNECT_TIMEOUT,
         max_time:        CURL_MAX_TIME,
+        # `Utils::Curl` uses curl's `--retry` option by default but this isn't
+        # valuable enough in the context of livecheck to justify the additional
+        # effort. A zero value ensures that the `--retry` option is omitted.
         retries:         0,
       }.freeze
 
diff --git a/Library/Homebrew/livecheck/strategy/github_latest.rb b/Library/Homebrew/livecheck/strategy/github_latest.rb
index 92792059b3895..8cb4bf67f800f 100644
--- a/Library/Homebrew/livecheck/strategy/github_latest.rb
+++ b/Library/Homebrew/livecheck/strategy/github_latest.rb
@@ -72,7 +72,7 @@ def self.generate_input_values(url)
           match = url.sub(/\.git$/i, "").match(URL_MATCH_REGEX)
           return values if match.blank?
 
-          values[:url] = "https://api.github.com/repos/#{match[:username]}/#{match[:repository]}/releases/latest"
+          values[:url] = "#{GitHub::API_URL}/repos/#{match[:username]}/#{match[:repository]}/releases/latest"
           values[:username] = match[:username]
           values[:repository] = match[:repository]
 
@@ -141,7 +141,11 @@ def self.find_versions(url:, regex: DEFAULT_REGEX, **_unused, &block)
 
           match_data[:url] = generated[:url]
 
-          release = GitHub.get_latest_release(generated[:username], generated[:repository])
+          release = GitHub.get_latest_release(
+            generated[:username],
+            generated[:repository],
+            **Strategy::DEFAULT_CURL_OPTIONS,
+          )
           versions_from_content(release, regex, &block).each do |match_text|
             match_data[:matches][match_text] = Version.new(match_text)
           end
diff --git a/Library/Homebrew/utils/github.rb b/Library/Homebrew/utils/github.rb
index 830c6c5a811c9..0efbbd9daed2b 100644
--- a/Library/Homebrew/utils/github.rb
+++ b/Library/Homebrew/utils/github.rb
@@ -248,9 +248,36 @@ def self.get_release(user, repo, tag)
     API.open_rest(url, request_method: :GET)
   end
 
-  def self.get_latest_release(user, repo)
+  sig {
+    params(
+      user:            String,
+      repo:            String,
+      debug:           T.nilable(T::Boolean),
+      timeout:         T.nilable(T.any(Integer, Float)),
+      connect_timeout: T.nilable(T.any(Integer, Float)),
+      max_time:        T.nilable(T.any(Integer, Float)),
+      retries:         T.nilable(Integer),
+    ).returns(T::Hash[String, T.untyped])
+  }
+  def self.get_latest_release(
+    user,
+    repo,
+    debug: nil,
+    timeout: nil,
+    connect_timeout: nil,
+    max_time: nil,
+    retries: nil
+  )
     url = "#{API_URL}/repos/#{user}/#{repo}/releases/latest"
-    API.open_rest(url, request_method: :GET)
+
+    curl_options = {
+      debug:           debug,
+      timeout:         timeout,
+      connect_timeout: connect_timeout,
+      max_time:        max_time,
+      retries:         retries,
+    }.compact
+    API.open_rest(url, request_method: :GET, **curl_options)
   end
 
   def self.generate_release_notes(user, repo, tag, previous_tag: nil)
diff --git a/Library/Homebrew/utils/github/api.rb b/Library/Homebrew/utils/github/api.rb
index d5e9bdb4ddfdd..787e923d76708 100644
--- a/Library/Homebrew/utils/github/api.rb
+++ b/Library/Homebrew/utils/github/api.rb
@@ -180,8 +180,33 @@ def self.credentials_error_message(response_headers, needed_scopes)
       EOS
     end
 
+    sig {
+      params(
+        url:              T.any(String, URI::HTTPS),
+        data:             T.nilable(T::Hash[String, T.untyped]),
+        data_binary_path: T.nilable(String),
+        request_method:   T.nilable(Symbol),
+        scopes:           Array,
+        parse_json:       T::Boolean,
+        debug:            T.nilable(T::Boolean),
+        timeout:          T.nilable(T.any(Integer, Float)),
+        connect_timeout:  T.nilable(T.any(Integer, Float)),
+        max_time:         T.nilable(T.any(Integer, Float)),
+        retries:          T.nilable(Integer),
+      ).returns(T.untyped)
+    }
     def self.open_rest(
-      url, data: nil, data_binary_path: nil, request_method: nil, scopes: [].freeze, parse_json: true
+      url,
+      data: nil,
+      data_binary_path: nil,
+      request_method: nil,
+      scopes: [].freeze,
+      parse_json: true,
+      debug: nil,
+      timeout: nil,
+      connect_timeout: nil,
+      max_time: nil,
+      retries: nil
     )
       # This is a no-op if the user is opting out of using the GitHub API.
       return block_given? ? yield({}) : {} if Homebrew::EnvConfig.no_github_api?
@@ -222,7 +247,15 @@ def self.open_rest(
 
         args += ["--dump-header", T.must(headers_tmpfile.path)]
 
-        output, errors, status = curl_output("--location", url.to_s, *args, secrets: [token])
+        curl_options = {
+          debug:           debug,
+          timeout:         timeout,
+          connect_timeout: connect_timeout,
+          max_time:        max_time,
+          retries:         retries,
+        }.compact
+
+        output, errors, status = curl_output("--location", url.to_s, *args, secrets: [token], **curl_options)
         output, _, http_code = output.rpartition("\n")
         output, _, http_code = output.rpartition("\n") if http_code == "000"
         headers = headers_tmpfile.read
@@ -262,7 +295,7 @@ def self.paginate_rest(url, additional_query_params: nil, per_page: 100)
 
     def self.open_graphql(query, variables: nil, scopes: [].freeze, raise_errors: true)
       data = { query: query, variables: variables }
-      result = open_rest("#{API_URL}/graphql", scopes: scopes, data: data, request_method: "POST")
+      result = open_rest("#{API_URL}/graphql", scopes: scopes, data: data, request_method: :POST)
 
       if raise_errors
         if result["errors"].present?